Beckmann Sample, near normal, slope_y

Average Accuracy: 57.4% → 98.3%

Time: 16.3s

Precision: binary32

Cost: 19584

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(-u1\right)\\ \sqrt{-\frac{{t_0}^{2}}{t_0}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

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
 (let* ((t_0 (log1p (- u1))))
   (* (sqrt (- (/ (pow t_0 2.0) t_0))) (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) {
	float t_0 = log1pf(-u1);
	return sqrtf(-(powf(t_0, 2.0f) / t_0)) * 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)
	t_0 = log1p(Float32(-u1))
	return Float32(sqrt(Float32(-Float32((t_0 ^ Float32(2.0)) / t_0))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

Derivation

1. Initial program 57.4%

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

2. Applied egg-rr 98.1%

   \[\leadsto \sqrt{-\color{blue}{\frac{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}{0 + {\left(\mathsf{log1p}\left(-u1\right)\right)}^{3}} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   Proof

   [Start] 57.4	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   add-sqr-sqrt [=>] 1.7	\[ \sqrt{-\color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{\log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sqrt-unprod [=>] 1.7	\[ \sqrt{-\color{blue}{\sqrt{\log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sqr-neg [<=] 1.7	\[ \sqrt{-\sqrt{\color{blue}{\left(-\log \left(1 - u1\right)\right) \cdot \left(-\log \left(1 - u1\right)\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sqrt-unprod [<=] 1.7	\[ \sqrt{-\color{blue}{\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   add-sqr-sqrt [<=] 1.7	\[ \sqrt{-\color{blue}{\left(-\log \left(1 - u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   neg-sub0 [=>] 1.7	\[ \sqrt{-\color{blue}{\left(0 - \log \left(1 - u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   metadata-eval [<=] 1.7	\[ \sqrt{-\left(\color{blue}{\log 1} - \log \left(1 - u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   flip-- [=>] -0.0	\[ \sqrt{-\color{blue}{\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\log 1 + \log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   add-sqr-sqrt [=>] -0.0	\[ \sqrt{-\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\log 1 + \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{\log \left(1 - u1\right)}}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sqrt-unprod [=>] 55.7	\[ \sqrt{-\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\log 1 + \color{blue}{\sqrt{\log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sqr-neg [<=] 55.7	\[ \sqrt{-\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\log 1 + \sqrt{\color{blue}{\left(-\log \left(1 - u1\right)\right) \cdot \left(-\log \left(1 - u1\right)\right)}}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sqrt-unprod [<=] 55.7	\[ \sqrt{-\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\log 1 + \color{blue}{\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   add-sqr-sqrt [<=] 55.7	\[ \sqrt{-\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\log 1 + \color{blue}{\left(-\log \left(1 - u1\right)\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [<=] 55.7	\[ \sqrt{-\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\color{blue}{\log 1 - \log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   flip3-- [=>] 55.7	\[ \sqrt{-\frac{\log 1 \cdot \log 1 - \log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}{\color{blue}{\frac{{\log 1}^{3} - {\log \left(1 - u1\right)}^{3}}{\log 1 \cdot \log 1 + \left(\log \left(1 - u1\right) \cdot \log \left(1 - u1\right) + \log 1 \cdot \log \left(1 - u1\right)\right)}}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.3%

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

   [Start] 98.1	\[ \sqrt{-\frac{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}{0 + {\left(\mathsf{log1p}\left(-u1\right)\right)}^{3}} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   +-lft-identity [=>] 98.1	\[ \sqrt{-\frac{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}{\color{blue}{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{3}}} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   unpow3 [=>] 98.2	\[ \sqrt{-\frac{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}{\color{blue}{\left(\mathsf{log1p}\left(-u1\right) \cdot \mathsf{log1p}\left(-u1\right)\right) \cdot \mathsf{log1p}\left(-u1\right)}} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   unpow2 [<=] 98.2	\[ \sqrt{-\frac{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}{\color{blue}{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2}} \cdot \mathsf{log1p}\left(-u1\right)} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   +-rgt-identity [<=] 98.2	\[ \sqrt{-\frac{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}{\color{blue}{\left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \mathsf{log1p}\left(-u1\right)} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   associate-/r* [=>] 98.3	\[ \sqrt{-\color{blue}{\frac{\frac{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}}{\mathsf{log1p}\left(-u1\right)}} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   *-inverses [=>] 98.3	\[ \sqrt{-\frac{\color{blue}{1}}{\mathsf{log1p}\left(-u1\right)} \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   associate-/r/ [<=] 98.3	\[ \sqrt{-\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(-u1\right)}{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   associate-/l* [<=] 98.3	\[ \sqrt{-\color{blue}{\frac{1 \cdot \left({\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0\right)}{\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   *-lft-identity [=>] 98.3	\[ \sqrt{-\frac{\color{blue}{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2} + 0}}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   +-rgt-identity [=>] 98.3	\[ \sqrt{-\frac{\color{blue}{{\left(\mathsf{log1p}\left(-u1\right)\right)}^{2}}}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 98.3%

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

Alternatives

Alternative 1
Accuracy	95.7%
Cost	13476

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)}\\ \end{array} \]

Alternative 2
Accuracy	94.4%
Cost	13348

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 3
Accuracy	90.5%
Cost	13220

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.02800000086426735:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Accuracy	84.1%
Cost	13156

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004900000058114529:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + \left(u1 \cdot u1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5
Accuracy	98.4%
Cost	13056

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

Alternative 6
Accuracy	74.7%
Cost	6784

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

Alternative 7
Accuracy	66.7%
Cost	6592

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

Alternative 8
Accuracy	66.7%
Cost	6592

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

Reproduce

herbie shell --seed 2023131 
(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))))