Beckmann Sample, near normal, slope_x

Average Accuracy: 57.1% → 99.0%

Time: 15.9s

Precision: binary32

Cost: 39008

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{1.5}\right)\right) \cdot \sqrt{t_0}\right)\right) \end{array} \]

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
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+
     0.5
     (-
      (* 0.5 (cos (* PI (+ u2 u2))))
      (* (expm1 (log1p (pow t_0 1.5))) (sqrt t_0)))))))

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) {
	float t_0 = sinf((((float) M_PI) * u2));
	return sqrtf(-log1pf(-u1)) * (0.5f + ((0.5f * cosf((((float) M_PI) * (u2 + u2)))) - (expm1f(log1pf(powf(t_0, 1.5f))) * sqrtf(t_0))));
}

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)
	t_0 = sin(Float32(Float32(pi) * u2))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(0.5) + Float32(Float32(Float32(0.5) * cos(Float32(Float32(pi) * Float32(u2 + u2)))) - Float32(expm1(log1p((t_0 ^ Float32(1.5)))) * sqrt(t_0)))))
end

Derivation

1. Initial program 57.1%

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

2. Simplified 99.0%

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

   [Start] 57.1	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 57.1	\[ \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 99.0	\[ \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   associate-*l* [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Applied egg-rr 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right)} \]
   Proof

   [Start] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]
   cos-2 [=>] 98.8	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\cos \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right) - \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)} \]
   sqr-cos-a [=>] 98.9	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)} - \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right) \]
   associate--l+ [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) - \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right)} \]
   count-2 [<=] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \color{blue}{\left(\pi \cdot u2 + \pi \cdot u2\right)} - \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right) \]
   distribute-lft-out [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \color{blue}{\left(\pi \cdot \left(u2 + u2\right)\right)} - \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right) \]
   pow2 [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \color{blue}{{\sin \left(\pi \cdot u2\right)}^{2}}\right)\right) \]

4. Applied egg-rr 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \color{blue}{\left(\sin \left(\pi \cdot u2\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}}\right)\right) \]
   Proof

   [Start] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]
   unpow2 [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \color{blue}{\sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)}\right)\right) \]
   add-sqr-sqrt [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \sin \left(\pi \cdot u2\right) \cdot \color{blue}{\left(\sqrt{\sin \left(\pi \cdot u2\right)} \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)}\right)\right) \]
   associate-*r* [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \color{blue}{\left(\sin \left(\pi \cdot u2\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}}\right)\right) \]

5. Applied egg-rr 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin \left(\pi \cdot u2\right)}^{1.5}\right)\right)} \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]
   Proof

   [Start] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \left(\sin \left(\pi \cdot u2\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]
   expm1-log1p-u [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot u2\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right)} \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]
   *-commutative [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{\sin \left(\pi \cdot u2\right)} \cdot \sin \left(\pi \cdot u2\right)}\right)\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]
   pow1/2 [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\sin \left(\pi \cdot u2\right)}^{0.5}} \cdot \sin \left(\pi \cdot u2\right)\right)\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]
   pow-plus [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\sin \left(\pi \cdot u2\right)}^{\left(0.5 + 1\right)}}\right)\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]
   metadata-eval [=>] 99.0	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin \left(\pi \cdot u2\right)}^{\color{blue}{1.5}}\right)\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]

6. Final simplification 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin \left(\pi \cdot u2\right)}^{1.5}\right)\right) \cdot \sqrt{\sin \left(\pi \cdot u2\right)}\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	19872

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) + \frac{\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) + -1}{2}\right)\right) \]

Alternative 2
Accuracy	96.3%
Cost	13476

\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.0005799999926239252:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(-0.5 - u1 \cdot 0.3333333333333333\right)} \cdot \cos t_0\\ \end{array} \]

Alternative 3
Accuracy	94.7%
Cost	13348

\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.0005799999926239252:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 4
Accuracy	90.8%
Cost	13156

\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5
Accuracy	99.0%
Cost	13056

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

Alternative 6
Accuracy	79.8%
Cost	6496

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

Alternative 7
Accuracy	76.5%
Cost	3680

\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)} \]

Alternative 8
Accuracy	75.3%
Cost	3552

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

Alternative 9
Accuracy	72.8%
Cost	3424

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

Alternative 10
Accuracy	64.8%
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023138 
(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))))