Beckmann Sample, near normal, slope_y

Average Error: 13.5 → 0.6

Time: 14.5s

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

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

↓

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 2.0 (* u2 PI))) (sqrt (- (log1p u1) (log1p (* u1 (- u1)))))))

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

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

Derivation

1. Initial program 13.5

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

2. Applied egg-rr 3.1

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

3. Simplified 0.6

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

   [Start] 3.1	\[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + -1 \cdot \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   mul-1-neg [=>] 3.1	\[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + \color{blue}{\left(-\mathsf{log1p}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [<=] 3.1	\[ \sqrt{-\color{blue}{\left(\log \left(1 - u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 3.1	\[ \sqrt{-\left(\log \color{blue}{\left(1 + \left(-u1 \cdot u1\right)\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 0.6	\[ \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(-u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   distribute-rgt-neg-in [=>] 0.6	\[ \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{u1 \cdot \left(-u1\right)}\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Taylor expanded in u2 around inf 14.4

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

5. Simplified 0.6

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

   [Start] 14.4	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]
   log1p-def [=>] 3.1	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]
   log1p-def [=>] 0.6	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(-1 \cdot {u1}^{2}\right)}} \]
   mul-1-neg [=>] 0.6	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{-{u1}^{2}}\right)} \]
   unpow2 [=>] 0.6	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(-\color{blue}{u1 \cdot u1}\right)} \]
   distribute-rgt-neg-in [=>] 0.6	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot \left(-u1\right)}\right)} \]

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	1.4
Cost	13476

\[\begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.0010000000474974513:\\ \;\;\;\;\left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \sin t_0\\ \end{array} \]

Alternative 2
Error	1.9
Cost	13348

\[\begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.004399999976158142:\\ \;\;\;\;\left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 3
Error	3.1
Cost	13220

\[\begin{array}{l} t_0 := 2 \cdot \left(u2 \cdot \pi\right)\\ \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.009999999776482582:\\ \;\;\;\;t_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	13056

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

Alternative 5
Error	5.2
Cost	9860

\[\begin{array}{l} t_0 := 2 \cdot \left(u2 \cdot \pi\right)\\ \mathbf{if}\;u2 \leq 0.0015999999595806003:\\ \;\;\;\;t_0 \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \]

Alternative 6
Error	8.4
Cost	6784

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

Alternative 7
Error	10.9
Cost	6592

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

Alternative 8
Error	10.9
Cost	6592

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

Alternative 9
Error	29.7
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023066 
(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))))