Beckmann Sample, near normal, slope_y

Average Error: 13.7 → 0.5

Time: 13.6s

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

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

↓

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

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
 (* (sqrt (- (log1p (- u1)))) (sin (* 2.0 (* PI 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) {
	return sqrtf(-log1pf(-u1)) * sinf((2.0f * (((float) M_PI) * u2)));
}

Julia

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(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))
end

Derivation

1. Initial program 13.7

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

2. Simplified 0.5

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

   [Start] 13.7	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 13.7	\[ \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 0.5	\[ \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   associate-*l* [=>] 0.5	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Final simplification 0.5

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

Alternatives

Alternative 1
Error	1.4
Cost	13476

\[\begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.00279999990016222:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \sin t_0\\ \end{array} \]

Alternative 2
Error	1.8
Cost	13348

\[\begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.00279999990016222:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \end{array} \]

Alternative 3
Error	2.9
Cost	13220

\[\begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004999999888241291:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	5.1
Cost	9860

\[\begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;u2 \leq 0.0007999999797903001:\\ \;\;\;\;t_0 \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5
Error	8.2
Cost	6784

\[\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)} \]

Alternative 6
Error	10.7
Cost	6592

\[2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{u1}\right)\right) \]

Alternative 7
Error	10.7
Cost	6592

\[\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1} \]

Alternative 8
Error	30.4
Cost	6560

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

Alternative 9
Error	30.6
Cost	6496

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

Alternative 10
Error	30.6
Cost	3296

\[\sqrt{u1} \cdot -2 \]

Reproduce

herbie shell --seed 2023016 
(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))))