Average Error: 13.5 → 0.3
Time: 17.2s
Precision: binary32
Cost: 22912
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (+ 0.5 (- (* 0.5 (cos (* 2.0 (* PI u2)))) (pow (sin (* PI u2)) 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) {
	return sqrtf(-log1pf(-u1)) * (0.5f + ((0.5f * cosf((2.0f * (((float) M_PI) * u2)))) - powf(sinf((((float) M_PI) * u2)), 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)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(0.5) + Float32(Float32(Float32(0.5) * cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2)))) - (sin(Float32(Float32(pi) * u2)) ^ Float32(2.0)))))
end

\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

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

  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)))): 227 points increase in error, 6 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)))): 0 points increase in error, 0 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, 0 points decrease in error

  3. Applied egg-rr0.4

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

  4. Applied egg-rr0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)} - {\sin \left(\pi \cdot u2\right)}^{2}\right) \]

  5. Applied egg-rr0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\left(0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right) + 0.5\right)} \]

  6. Final simplification0.3

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]

Alternatives

Alternative 1
Error3.0
Cost16356
\[\begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999600052833557:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
Alternative 2
Error0.3
Cost16352
\[\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(u1 \cdot \left(-u1\right)\right)} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \]
Alternative 3
Error1.7
Cost13348
\[\begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.0011500000255182385:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \]
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
\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \]
Alternative 6
Error8.7
Cost6560
\[\sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)} \]
Alternative 7
Error6.3
Cost6496
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \]
Alternative 8
Error11.2
Cost3232
\[\sqrt{u1} \]

Error

Reproduce

herbie shell --seed 2022308 
(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))))