Beckmann Sample, near normal, slope_x

Average Error: 13.6 → 0.4

Time: 15.3s

Precision: binary32

Cost: 16352

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

Math

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

↓

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

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
 (* (cos (* (* 2.0 PI) u2)) (sqrt (- (log1p u1) (log1p (* u1 (- 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) - log1pf((u1 * -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(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(log1p(u1) - log1p(Float32(u1 * Float32(-u1))))))
end

Derivation

1. Initial program 13.6

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

2. Applied egg-rr 3.0

   \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - u1 \cdot u1\right) + -1 \cdot \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 0.4

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

   [Start] 3.0	\[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + -1 \cdot \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   mul-1-neg [=>] 3.0	\[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + \color{blue}{\left(-\mathsf{log1p}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [<=] 3.0	\[ \sqrt{-\color{blue}{\left(\log \left(1 - u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 3.0	\[ \sqrt{-\left(\log \color{blue}{\left(1 + \left(-u1 \cdot u1\right)\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 0.4	\[ \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(-u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   distribute-rgt-neg-in [=>] 0.4	\[ \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{u1 \cdot \left(-u1\right)}\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.3
Cost	16676

\[\begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999995231628418:\\ \;\;\;\;t_0 \cdot \sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 + u1 \cdot -0.3333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(u1 \cdot \left(-u1\right)\right)}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	13412

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(u1 \cdot \left(-u1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{2 \cdot \left(u1 \cdot \left(0.5 + u1 \cdot 0.25\right)\right)}\\ \end{array} \]

Alternative 3
Error	3.1
Cost	13156

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(u1 \cdot \left(-u1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	13056

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

Alternative 5
Error	6.7
Cost	9792

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

Alternative 6
Error	6.7
Cost	6496

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

Alternative 7
Error	7.7
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 8
Error	8.1
Cost	3552

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

Alternative 9
Error	8.9
Cost	3424

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

Alternative 10
Error	11.4
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023066 
(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))))