Beckmann Sample, near normal, slope_x

Average Accuracy: 57.8% → 99.0%

Time: 12.9s

Precision: binary32

Cost: 32576

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

Math

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

↓

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) + \left(0.5 - {\sin \left(\sqrt[3]{{\pi}^{3} \cdot {u2}^{3}}\right)}^{2}\right)\right) \]

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
 (*
  (sqrt (- (log1p (- u1))))
  (+
   (* 0.5 (cos (* PI (+ u2 u2))))
   (- 0.5 (pow (sin (cbrt (* (pow PI 3.0) (pow u2 3.0)))) 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) {
	return sqrtf(-log1pf(-u1)) * ((0.5f * cosf((((float) M_PI) * (u2 + u2)))) + (0.5f - powf(sinf(cbrtf((powf(((float) M_PI), 3.0f) * powf(u2, 3.0f)))), 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)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(Float32(0.5) * cos(Float32(Float32(pi) * Float32(u2 + u2)))) + Float32(Float32(0.5) - (sin(cbrt(Float32((Float32(pi) ^ Float32(3.0)) * (u2 ^ Float32(3.0))))) ^ Float32(2.0)))))
end

Derivation

1. Initial program 57.8%

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

2. Simplified 99.0%

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

   [Start] 57.8	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 57.8	\[ \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 99.0	\[ \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   associate-*l* [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Applied egg-rr 99.0%

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

4. Simplified 99.0%

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

   [Start] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) + \left(-{\sin \left(\pi \cdot u2\right)}^{2}\right)\right)\right) \]
   +-commutative [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) + \left(-{\sin \left(\pi \cdot u2\right)}^{2}\right)\right) + 0.5\right)} \]
   sub-neg [<=] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)} + 0.5\right) \]
   associate-+l- [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right)} \]
   *-commutative [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \left({\sin \color{blue}{\left(u2 \cdot \pi\right)}}^{2} - 0.5\right)\right) \]

5. Applied egg-rr 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \left({\sin \color{blue}{\left(\sqrt[3]{{\pi}^{3} \cdot {u2}^{3}}\right)}}^{2} - 0.5\right)\right) \]

6. Final simplification 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) + \left(0.5 - {\sin \left(\sqrt[3]{{\pi}^{3} \cdot {u2}^{3}}\right)}^{2}\right)\right) \]

Alternatives

Alternative 1
Accuracy	94.9%
Cost	16548

\[\begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999989867210388:\\ \;\;\;\;t_0 \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 2
Accuracy	90.8%
Cost	13156

\[\begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3
Accuracy	99.0%
Cost	13056

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

Alternative 4
Accuracy	95.6%
Cost	10180

\[\begin{array}{l} \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 5
Accuracy	79.5%
Cost	6496

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

Alternative 6
Accuracy	76.1%
Cost	3680

\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot \left(u1 \cdot 0.25 + 0.3333333333333333\right)\right)} \]

Alternative 7
Accuracy	74.8%
Cost	3552

\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \]

Alternative 8
Accuracy	72.2%
Cost	3424

\[\sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)} \]

Alternative 9
Accuracy	64.3%
Cost	3232

\[\sqrt{u1} \]

Alternative 10
Accuracy	-0.0%
Cost	96

\[\frac{-1}{0} \]

Reproduce

herbie shell --seed 2023122 
(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))))