Average Error: 13.6 → 0.5
Time: 11.3s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\pi \cdot 4\right)\right)\right) \cdot \left(\pi \cdot 2\right)\right) \cdot {u2}^{3}\right)\right)}\right) \]
(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
   (cbrt
    (expm1
     (log1p
      (* (* (expm1 (log1p (* PI (* PI 4.0)))) (* 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(cbrtf(expm1f(log1pf(((expm1f(log1pf((((float) M_PI) * (((float) M_PI) * 4.0f)))) * (((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(cbrt(expm1(log1p(Float32(Float32(expm1(log1p(Float32(Float32(pi) * Float32(Float32(pi) * Float32(4.0))))) * Float32(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(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\pi \cdot 4\right)\right)\right) \cdot \left(\pi \cdot 2\right)\right) \cdot {u2}^{3}\right)\right)}\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 13.6

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

  2. Simplified0.5

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

  3. Applied add-cbrt-cube_binary320.5

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

  4. Applied add-cbrt-cube_binary320.5

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

  5. Applied cbrt-unprod_binary320.5

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

  6. Applied expm1-log1p-u_binary320.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot \left(2 \cdot \pi\right)\right)\right)} \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

  7. Simplified0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\pi \cdot \left(\pi \cdot 4\right)\right)}\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

  8. Taylor expanded in u2 around 0 0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\pi \cdot 4\right)\right)\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{u2}^{3}}}\right) \]

  9. Applied expm1-log1p-u_binary320.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\pi \cdot 4\right)\right)\right) \cdot \left(2 \cdot \pi\right)\right) \cdot {u2}^{3}\right)\right)}}\right) \]

  10. Final simplification0.5

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\pi \cdot 4\right)\right)\right) \cdot \left(\pi \cdot 2\right)\right) \cdot {u2}^{3}\right)\right)}\right) \]

Reproduce

herbie shell --seed 2022131 
(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))))