Beckmann Sample, near normal, slope_x

Average Accuracy: 57.2% → 99.0%

Time: 15.7s

Precision: binary32

Cost: 13056

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

Math

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

↓

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

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
 (* (cos (* 2.0 (* PI u2))) (sqrt (- (log1p (- u1))))))

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 cosf((2.0f * (((float) M_PI) * u2))) * sqrtf(-log1pf(-u1));
}

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(cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(Float32(-log1p(Float32(-u1)))))
end

Derivation

1. Initial program 57.2%

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

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] 57.2	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 57.2	\[ \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. Final simplification 99.0%

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

Alternatives

Alternative 1
Accuracy	95.4%
Cost	13476

\[\begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.05999999865889549:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 + u1 \cdot -0.3333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(u1 \cdot \left(-u1\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	94.7%
Cost	13348

\[\begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.0026000000070780516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 - \left(u1 \cdot u1\right) \cdot -0.5}\\ \end{array} \]

Alternative 3
Accuracy	90.7%
Cost	13156

\[\begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.007000000216066837:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	79.9%
Cost	6496

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

Alternative 5
Accuracy	75.3%
Cost	3552

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

Alternative 6
Accuracy	72.7%
Cost	3424

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

Alternative 7
Accuracy	64.9%
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023151 
(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))))