Beckmann Sample, near normal, slope_y

Average Error: 13.5 → 0.5

Time: 18.2s

Precision: binary32

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := -\mathsf{log1p}\left(-u1\right)\\ t_1 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \sqrt[3]{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right) \cdot \sqrt{t_0}\right) \cdot \left(\left(t_1 \cdot t_1\right) \cdot \sin \left(\sqrt{u2} \cdot \left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log1p (- u1)))) (t_1 (sin (* (* 2.0 PI) u2))))
   (cbrt
    (*
     (* (log1p (expm1 t_0)) (sqrt t_0))
     (* (* t_1 t_1) (sin (* (sqrt u2) (* (* 2.0 PI) (sqrt u2)))))))))

C

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) {
	float t_0 = -log1pf(-u1);
	float t_1 = sinf((2.0f * ((float) M_PI)) * u2);
	return cbrtf((log1pf(expm1f(t_0)) * sqrtf(t_0)) * ((t_1 * t_1) * sinf(sqrtf(u2) * ((2.0f * ((float) M_PI)) * sqrtf(u2)))));
}

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Derivation

1. Initial program 13.5

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

2. Simplified 0.5

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

3. Applied add-cbrt-cube_binary32 0.5

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

4. Applied add-cbrt-cube_binary32 0.5

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

5. Applied cbrt-unprod_binary32 0.6

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

6. Applied add-sqr-sqrt_binary32 0.6

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

7. Applied associate-*r*_binary32 0.6

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

8. Applied log1p-expm1-u_binary32 0.6

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

9. Simplified 0.5

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

10. Final simplification 0.5

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

Reproduce

herbie shell --seed 2022068 
(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))))