Beckmann Sample, near normal, slope_y

Average Error: 13.8 → 0.5

Time: 15.1s

Precision: binary32

Cost: 22784

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

Math

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

↓

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

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 (* PI (* 2.0 u2)))
  (sqrt (- (log1p (fma u1 u1 u1)) (log1p (- (pow u1 3.0)))))))

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((((float) M_PI) * (2.0f * u2))) * sqrtf((log1pf(fmaf(u1, u1, u1)) - log1pf(-powf(u1, 3.0f))));
}

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(pi) * Float32(Float32(2.0) * u2))) * sqrt(Float32(log1p(fma(u1, u1, u1)) - log1p(Float32(-(u1 ^ Float32(3.0)))))))
end

Derivation

1. Initial program 13.8

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

2. Applied egg-rr 14.7

   \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 - {u1}^{3}\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 0.5

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

   [Start] 14.7	\[ \sqrt{-\left(\log \left(1 - {u1}^{3}\right) - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 14.7	\[ \sqrt{-\left(\log \color{blue}{\left(1 + \left(-{u1}^{3}\right)\right)} - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 14.2	\[ \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(-{u1}^{3}\right)} - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 0.5	\[ \sqrt{-\left(\mathsf{log1p}\left(-{u1}^{3}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Taylor expanded in u2 around inf 14.8

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\log \left({u1}^{2} + \left(1 + u1\right)\right) - \log \left(1 - {u1}^{3}\right)}} \]

5. Simplified 0.5

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

   [Start] 14.8	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\log \left({u1}^{2} + \left(1 + u1\right)\right) - \log \left(1 - {u1}^{3}\right)} \]
   associate-*r* [=>] 14.8	\[ \sin \color{blue}{\left(\left(2 \cdot u2\right) \cdot \pi\right)} \cdot \sqrt{\log \left({u1}^{2} + \left(1 + u1\right)\right) - \log \left(1 - {u1}^{3}\right)} \]
   *-commutative [=>] 14.8	\[ \sin \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \cdot \sqrt{\log \left({u1}^{2} + \left(1 + u1\right)\right) - \log \left(1 - {u1}^{3}\right)} \]
   associate-+r+ [=>] 14.7	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\log \color{blue}{\left(\left({u1}^{2} + 1\right) + u1\right)} - \log \left(1 - {u1}^{3}\right)} \]
   +-commutative [=>] 14.7	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\log \left(\color{blue}{\left(1 + {u1}^{2}\right)} + u1\right) - \log \left(1 - {u1}^{3}\right)} \]
   associate-+r+ [<=] 14.7	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\log \color{blue}{\left(1 + \left({u1}^{2} + u1\right)\right)} - \log \left(1 - {u1}^{3}\right)} \]
   unpow2 [=>] 14.7	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\log \left(1 + \left(\color{blue}{u1 \cdot u1} + u1\right)\right) - \log \left(1 - {u1}^{3}\right)} \]
   fma-udef [<=] 14.7	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\log \left(1 + \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right) - \log \left(1 - {u1}^{3}\right)} \]
   log1p-def [=>] 1.1	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)} - \log \left(1 - {u1}^{3}\right)} \]
   sub-neg [=>] 1.1	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right) - \log \color{blue}{\left(1 + \left(-{u1}^{3}\right)\right)}} \]
   log1p-def [=>] 0.5	\[ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right) - \color{blue}{\mathsf{log1p}\left(-{u1}^{3}\right)}} \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	1.8
Cost	13348

\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \end{array} \]

Alternative 2
Error	2.9
Cost	13220

\[\begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	13056

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

Alternative 4
Error	1.6
Cost	10180

\[\begin{array}{l} \mathbf{if}\;u1 \leq 0.015200000256299973:\\ \;\;\;\;\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]

Alternative 5
Error	11.0
Cost	9792

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

Alternative 6
Error	7.4
Cost	9792

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

Alternative 7
Error	29.7
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023045 
(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))))