Average Error: 13.8 → 0.3
Time: 14.1s
Precision: binary32
Cost: 39008
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) + 2 \cdot \mathsf{fma}\left(-t_0, t_0, {t_0}^{2}\right)\right) \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+ (cos (* PI (+ u2 u2))) (* 2.0 (fma (- t_0) t_0 (pow t_0 2.0)))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}
↓
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((((float) M_PI) * u2));
	return sqrtf(-log1pf(-u1)) * (cosf((((float) M_PI) * (u2 + u2))) + (2.0f * fmaf(-t_0, t_0, powf(t_0, 2.0f))));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(pi) * u2))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) + Float32(Float32(2.0) * fma(Float32(-t_0), t_0, (t_0 ^ Float32(2.0))))))
end

\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\begin{array}{l}
t_0 := \sin \left(\pi \cdot u2\right)\\
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) + 2 \cdot \mathsf{fma}\left(-t_0, t_0, {t_0}^{2}\right)\right)
\end{array}

Error

Derivation

  1. Initial program 13.8

    \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  2. Simplified0.3

    \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
    Proof
    (*.f32 (sqrt.f32 (neg.f32 (log1p.f32 (neg.f32 u1)))) (cos.f32 (*.f32 2 (*.f32 (PI.f32) u2)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (sqrt.f32 (neg.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 u1)))))) (cos.f32 (*.f32 2 (*.f32 (PI.f32) u2)))): 1 points increase in error, 3 points decrease in error
    (*.f32 (sqrt.f32 (neg.f32 (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 u1))))) (cos.f32 (*.f32 2 (*.f32 (PI.f32) u2)))): 3 points increase in error, 1 points decrease in error
    (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 1 u1)))) (cos.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 2 (PI.f32)) u2)))): 0 points increase in error, 3 points decrease in error

  3. Applied egg-rr0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) + \left(\mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \left(\pi \cdot u2\right)}^{2}\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \left(\pi \cdot u2\right)}^{2}\right)\right)\right)} \]

  4. Simplified0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) + 2 \cdot \mathsf{fma}\left(-\sin \left(u2 \cdot \pi\right), \sin \left(u2 \cdot \pi\right), {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right)} \]
    Proof
    (*.f32 (sqrt.f32 (neg.f32 (log1p.f32 (neg.f32 u1)))) (+.f32 (cos.f32 (*.f32 (PI.f32) (+.f32 u2 u2))) (*.f32 2 (fma.f32 (neg.f32 (sin.f32 (*.f32 u2 (PI.f32)))) (sin.f32 (*.f32 u2 (PI.f32))) (pow.f32 (sin.f32 (*.f32 u2 (PI.f32))) 2))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (sqrt.f32 (neg.f32 (log1p.f32 (neg.f32 u1)))) (+.f32 (cos.f32 (*.f32 (PI.f32) (+.f32 u2 u2))) (*.f32 2 (fma.f32 (neg.f32 (sin.f32 (Rewrite<= *-commutative_binary32 (*.f32 (PI.f32) u2)))) (sin.f32 (*.f32 u2 (PI.f32))) (pow.f32 (sin.f32 (*.f32 u2 (PI.f32))) 2))))): 5 points increase in error, 0 points decrease in error
    (*.f32 (sqrt.f32 (neg.f32 (log1p.f32 (neg.f32 u1)))) (+.f32 (cos.f32 (*.f32 (PI.f32) (+.f32 u2 u2))) (*.f32 2 (fma.f32 (neg.f32 (sin.f32 (*.f32 (PI.f32) u2))) (sin.f32 (Rewrite<= *-commutative_binary32 (*.f32 (PI.f32) u2))) (pow.f32 (sin.f32 (*.f32 u2 (PI.f32))) 2))))): 0 points increase in error, 5 points decrease in error
    (*.f32 (sqrt.f32 (neg.f32 (log1p.f32 (neg.f32 u1)))) (+.f32 (cos.f32 (*.f32 (PI.f32) (+.f32 u2 u2))) (*.f32 2 (fma.f32 (neg.f32 (sin.f32 (*.f32 (PI.f32) u2))) (sin.f32 (*.f32 (PI.f32) u2)) (pow.f32 (sin.f32 (Rewrite<= *-commutative_binary32 (*.f32 (PI.f32) u2))) 2))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (sqrt.f32 (neg.f32 (log1p.f32 (neg.f32 u1)))) (+.f32 (cos.f32 (*.f32 (PI.f32) (+.f32 u2 u2))) (Rewrite<= count-2_binary32 (+.f32 (fma.f32 (neg.f32 (sin.f32 (*.f32 (PI.f32) u2))) (sin.f32 (*.f32 (PI.f32) u2)) (pow.f32 (sin.f32 (*.f32 (PI.f32) u2)) 2)) (fma.f32 (neg.f32 (sin.f32 (*.f32 (PI.f32) u2))) (sin.f32 (*.f32 (PI.f32) u2)) (pow.f32 (sin.f32 (*.f32 (PI.f32) u2)) 2)))))): 0 points increase in error, 0 points decrease in error

  5. Final simplification0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) + 2 \cdot \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]

Alternatives

Alternative 1
Error0.3
Cost22912
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot 0.5 + \left(0.5 - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]
Alternative 2
Error2.9
Cost16356
\[\begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \leq 0.9999970197677612:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
Alternative 3
Error1.8
Cost13348
\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.0003000000142492354:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \end{array} \]
Alternative 4
Error0.3
Cost13056
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]
Alternative 5
Error2.6
Cost10112
\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \]
Alternative 6
Error6.3
Cost6496
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \]
Alternative 7
Error7.5
Cost3680
\[\sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot \left(u1 \cdot 0.25 + 0.3333333333333333\right)\right)} \]
Alternative 8
Error7.9
Cost3552
\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \]
Alternative 9
Error8.7
Cost3424
\[\sqrt{u1 - \left(u1 \cdot u1\right) \cdot -0.5} \]
Alternative 10
Error11.2
Cost3232
\[\sqrt{u1} \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))