Beckmann Sample, near normal, slope_x

Average Error: 13.5 → 0.3

Time: 14.8s

Precision: binary32

Cost: 45344

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) + \mathsf{fma}\left(-t_0, t_0, {\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot u2\right)\right)\right)}^{2}\right)\right) \end{array} \]

FPCore

(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 2.0)))
     (fma (- t_0) t_0 (pow (sin (expm1 (log1p (* PI u2)))) 2.0))))))

C

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 * 2.0f))) + fmaf(-t_0, t_0, powf(sinf(expm1f(log1pf((((float) M_PI) * u2)))), 2.0f)));
}

Julia

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 * Float32(2.0)))) + fma(Float32(-t_0), t_0, (sin(expm1(log1p(Float32(Float32(pi) * u2)))) ^ Float32(2.0)))))
end

Derivation

1. Initial program 13.5

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
   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, 4 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-rr 0.3

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

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot u2\right)\right)\right)}^{2}\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	38944

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

Alternative 2
Error	0.3
Cost	19456

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot e^{\log \left(\pi \cdot u2\right)}\right) \]

Alternative 3
Error	0.3
Cost	16384

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\mathsf{fma}\left(u2, \pi, -1\right) + \frac{u2}{u2}\right)\right) \]

Alternative 4
Error	3.1
Cost	16356

\[\begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \leq 0.9999279975891113:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	13348

\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.0010999999940395355:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \cdot \cos t_0\\ \end{array} \]

Alternative 6
Error	0.3
Cost	13056

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

Alternative 7
Error	6.4
Cost	6496

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \]

Alternative 8
Error	11.2
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2022325 
(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))))