Average Error: 13.4 → 0.5
Time: 5.3s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \sqrt[3]{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot {\pi}^{2}\right)\right) \cdot {u2}^{3}}\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (sin
   (*
    2.0
    (cbrt
     (* (* (pow (cbrt PI) 2.0) (* (cbrt PI) (pow PI 2.0))) (pow u2 3.0)))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}
↓
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((2.0f * cbrtf(((powf(cbrtf(((float) M_PI)), 2.0f) * (cbrtf(((float) M_PI)) * powf(((float) M_PI), 2.0f))) * powf(u2, 3.0f)))));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(2.0) * cbrt(Float32(Float32((cbrt(Float32(pi)) ^ Float32(2.0)) * Float32(cbrt(Float32(pi)) * (Float32(pi) ^ Float32(2.0)))) * (u2 ^ Float32(3.0)))))))
end

\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \sqrt[3]{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot {\pi}^{2}\right)\right) \cdot {u2}^{3}}\right)

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.4

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

  2. Simplified0.5

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

  3. Applied egg-rr0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \color{blue}{\sqrt[3]{{\pi}^{3} \cdot {u2}^{3}}}\right) \]

  4. Applied egg-rr0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \sqrt[3]{\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot {\pi}^{2}\right)\right)} \cdot {u2}^{3}}\right) \]

  5. Final simplification0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \sqrt[3]{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot {\pi}^{2}\right)\right) \cdot {u2}^{3}}\right) \]

Reproduce

herbie shell --seed 2022160 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 PI) u2))))