demo-atan2-02 false

category branch-cuts

Summary

Given

\(x, y \in \mathbb{R},\quad x < 0,\quad y > 0\)

Claim

\(\displaystyle \operatorname{atan2}\left(y, x\right) \neq \arctan\left(\frac{y}{x}\right)\)

Expected

false unequal at every admissible point (e.g. at \(y = 1, x = -1\))

the two sides differ at every admissible point

witness (false): \(y = 1, x = -1\)

Answers (latest run per system; open code for the reproducer that ran)

system outcome answer steps (from the trace) s run query (as sent) detail

none

always-none no verdict null 0.0 2026-10-13 17:20:05 df65d778fc5c (real x) ∧ (real y) ∧ (negative x) ∧ (positive y) ⊢ (= (atan2 y x) (atan (/ y x))) null

sympy

sympy-ask no verdict null difference → -atan(y/x) + atan2(y, x) · ask → None 0.112 2026-10-13 17:20:46 df65d778fc5c Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
y = Symbol('y', real=True)
Assum = And(Q.real(x), Q.real(y), Q.negative(x), Q.positive(y))
Residual = atan2(y, x) - atan(y / x)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`-atan(y/x) + atan2(y, x)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -atan(y/x) + atan2(y, x) · ask → None 0.03 2026-10-12 20:21:28 d7af4def9aaa Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
y = Symbol('y', real=True)
Assum = And(Q.real(x), Q.real(y), Q.negative(x), Q.positive(y))
Residual = atan2(y, x) - atan(y / x)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`-atan(y/x) + atan2(y, x)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine solved false difference → -atan(y/x) + atan2(y, x) · refine → pi 0.047 2026-10-13 17:20:49 df65d778fc5c Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) false

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
y = Symbol('y', real=True)
Assum = And(Q.real(x), Q.real(y), Q.negative(x), Q.positive(y))
Residual = atan2(y, x) - atan(y / x)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`-atan(y/x) + atan2(y, x)`
refine	`refine(Residual, Assum)`	→	`pi`
answer			`false`

sympy-refine solved false difference → -atan(y/x) + atan2(y, x) · refine → pi 0.044 2026-10-12 20:21:30 d7af4def9aaa Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) false

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
y = Symbol('y', real=True)
Assum = And(Q.real(x), Q.real(y), Q.negative(x), Q.positive(y))
Residual = atan2(y, x) - atan(y / x)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`-atan(y/x) + atan2(y, x)`
refine	`refine(Residual, Assum)`	→	`pi`
answer			`false`

sympy-simplify solved false difference → pi · simplify → pi 0.007 2026-10-13 17:20:51 df65d778fc5c Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) false

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True, negative=True)
y = Symbol('y', real=True, positive=True)
Residual = atan2(y, x) - atan(y / x)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`pi`
simplify	`simplify(Residual)`	→	`pi`
answer			`false`

sympy-simplify solved false difference → pi · simplify → pi 0.014 2026-10-12 20:21:32 d7af4def9aaa Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) false

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True, negative=True)
y = Symbol('y', real=True, positive=True)
Residual = atan2(y, x) - atan(y / x)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`pi`
simplify	`simplify(Residual)`	→	`pi`
answer			`false`

wolfram

wolfram-fullsimplify solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.022 2026-10-13 17:20:59 df65d778fc5c Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`FullSimplify[Residual, Assum]`	→	`Pi`
answer			`false`

wolfram-refine solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.022 2026-10-13 17:20:56 df65d778fc5c Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`Refine[Residual, Assum]`	→	`Pi`
answer			`false`

wolfram-simplify solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.023 2026-10-13 17:20:52 df65d778fc5c Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`Simplify[Residual, Assum]`	→	`Pi`
answer			`false`

Older runs (11)

system outcome answer steps (from the trace) s run query (as sent) detail

wolfram-fullsimplify solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.023 2026-10-13 16:39:09 03f5149718d0 Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`FullSimplify[Residual, Assum]`	→	`Pi`
answer			`false`

wolfram-refine solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.022 2026-10-13 16:39:06 03f5149718d0 Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`Refine[Residual, Assum]`	→	`Pi`
answer			`false`

wolfram-simplify solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.023 2026-10-13 16:39:02 03f5149718d0 Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`Simplify[Residual, Assum]`	→	`Pi`
answer			`false`

sympy-simplify solved false difference → pi · simplify → pi 0.013 2026-10-13 16:39:01 03f5149718d0 Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) false

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True, negative=True)
y = Symbol('y', real=True, positive=True)
Residual = atan2(y, x) - atan(y / x)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`pi`
simplify	`simplify(Residual)`	→	`pi`
answer			`false`

sympy-refine solved false difference → -atan(y/x) + atan2(y, x) · refine → pi 0.046 2026-10-13 16:38:59 03f5149718d0 Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) false

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
y = Symbol('y', real=True)
Assum = And(Q.real(x), Q.real(y), Q.negative(x), Q.positive(y))
Residual = atan2(y, x) - atan(y / x)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`-atan(y/x) + atan2(y, x)`
refine	`refine(Residual, Assum)`	→	`pi`
answer			`false`

always-none no verdict null 0.0 2026-10-13 16:38:56 03f5149718d0 (real x) ∧ (real y) ∧ (negative x) ∧ (positive y) ⊢ (= (atan2 y x) (atan (/ y x))) null

sympy-ask no verdict null difference → -atan(y/x) + atan2(y, x) · ask → None 0.112 2026-10-13 16:38:56 03f5149718d0 Q.real(x) ∧ Q.real(y) ∧ Q.negative(x) ∧ Q.positive(y) ⊢ Eq(atan2(y, x), atan((y/x))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
y = Symbol('y', real=True)
Assum = And(Q.real(x), Q.real(y), Q.negative(x), Q.positive(y))
Residual = atan2(y, x) - atan(y / x)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`atan2(y, x) - atan(y / x)`	→	`-atan(y/x) + atan2(y, x)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.023 2026-10-12 20:21:41 d7af4def9aaa Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`FullSimplify[Residual, Assum]`	→	`Pi`
answer			`false`

wolfram-refine solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.022 2026-10-12 20:21:37 d7af4def9aaa Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`Refine[Residual, Assum]`	→	`Pi`
answer			`false`

wolfram-simplify solved false difference → -ArcTan[y/x] + ArcTan[x, y] · result → Pi 0.023 2026-10-12 20:21:34 d7af4def9aaa Element[x, Reals] ∧ Element[y, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (Element[y, Reals] && y > 0) ⊢ (ArcTan[x, y] == ArcTan[(y/x)]) false

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && (Element[y, Reals]) && ((Element[x, Reals] && x < 0)) && ((Element[y, Reals] && y > 0));
  Residual = (ArcTan[x, y]) - (ArcTan[y/x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[x, y]) - (ArcTan[y/x])`	→	`-ArcTan[y/x] + ArcTan[x, y]`
result	`Simplify[Residual, Assum]`	→	`Pi`
answer			`false`

always-none no verdict null 0.0 2026-10-12 20:21:33 d7af4def9aaa (real x) ∧ (real y) ∧ (negative x) ∧ (positive y) ⊢ (= (atan2 y x) (atan (/ y x))) null

Source (branch.sexp)

(case demo-atan2-02 (category branch-cuts)
  (var x real) (var y real)
  (assume (negative x)) (assume (positive y))
  (ask (= (atan2 y x) (atan (/ y x))))
  (expect false)
  (witness false (y 1) (x -1)))