Beckmann Sample, near normal, slope_x

Average Error: 13.6 → 0.3

Time: 14.1s

Precision: binary32

Cost: 29312

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

Math

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

↓

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right) + \left(0.5 - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

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
 (*
  (sqrt (- (log1p (- u1))))
  (+
   (* 0.5 (cos (expm1 (log1p (* u2 (* PI 2.0))))))
   (- 0.5 (pow (sin (* u2 PI)) 2.0)))))

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) {
	return sqrtf(-log1pf(-u1)) * ((0.5f * cosf(expm1f(log1pf((u2 * (((float) M_PI) * 2.0f)))))) + (0.5f - powf(sinf((u2 * ((float) M_PI))), 2.0f)));
}

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(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(Float32(0.5) * cos(expm1(log1p(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))))) + Float32(Float32(0.5) - (sin(Float32(u2 * Float32(pi))) ^ Float32(2.0)))))
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. Simplified 0.3

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

   [Start] 13.6	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 13.6	\[ \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 0.3	\[ \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   associate-*l* [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Applied egg-rr 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) + \left(-{\sin \left(\pi \cdot u2\right)}^{2}\right)\right)\right) \]
   +-commutative [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) + \left(-{\sin \left(\pi \cdot u2\right)}^{2}\right)\right) + 0.5\right)} \]
   sub-neg [<=] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - {\sin \left(\pi \cdot u2\right)}^{2}\right)} + 0.5\right) \]
   associate-+l- [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right)} \]
   *-commutative [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \color{blue}{\left(\left(u2 + u2\right) \cdot \pi\right)} - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right) \]
   count-2 [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\color{blue}{\left(2 \cdot u2\right)} \cdot \pi\right) - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right) \]
   associate-*r* [<=] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right) \]
   *-commutative [<=] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right) - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right) \]
   *-commutative [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \color{blue}{\left(\left(\pi \cdot u2\right) \cdot 2\right)} - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right) \]
   *-commutative [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\color{blue}{\left(u2 \cdot \pi\right)} \cdot 2\right) - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right) \]
   associate-*l* [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \color{blue}{\left(u2 \cdot \left(\pi \cdot 2\right)\right)} - \left({\sin \left(\pi \cdot u2\right)}^{2} - 0.5\right)\right) \]
   *-commutative [=>] 0.3	\[ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) - \left({\sin \color{blue}{\left(u2 \cdot \pi\right)}}^{2} - 0.5\right)\right) \]

5. Applied egg-rr 0.3

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)} - \left({\sin \left(u2 \cdot \pi\right)}^{2} - 0.5\right)\right) \]

6. Final simplification 0.3

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right) + \left(0.5 - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	23040

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

Alternative 2
Error	0.3
Cost	22912

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

Alternative 3
Error	0.3
Cost	22720

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

Alternative 4
Error	0.3
Cost	20000

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

Alternative 5
Error	2.9
Cost	16356

\[\begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \leq 0.9999799728393555:\\ \;\;\;\;\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 6
Error	1.8
Cost	13412

\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{2 \cdot \left(u1 \cdot \left(0.5 + u1 \cdot 0.25\right)\right)}\\ \end{array} \]

Alternative 7
Error	0.3
Cost	13056

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

Alternative 8
Error	2.5
Cost	10112

\[\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \]

Alternative 9
Error	6.4
Cost	6496

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

Alternative 10
Error	7.6
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 11
Error	8.0
Cost	3552

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

Alternative 12
Error	8.7
Cost	3424

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

Alternative 13
Error	11.3
Cost	3232

\[\sqrt{u1} \]

Alternative 14
Error	32.0
Cost	96

\[\frac{1}{0} \]

Reproduce

herbie shell --seed 2023025 
(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))))