Beckmann Sample, near normal, slope_x

Average Error: 13.6 → 0.3

Time: 14.6s

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

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

↓

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

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) (log1p (* u1 (- u1))))) (cos (* 2.0 (* u2 PI)))))

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

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(u1) - log1p(Float32(u1 * Float32(-u1))))) * cos(Float32(Float32(2.0) * Float32(u2 * Float32(pi)))))
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. Applied egg-rr 3.0

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

3. Simplified 0.3

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

   [Start] 3.0	\[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + -1 \cdot \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   mul-1-neg [=>] 3.0	\[ \sqrt{-\left(\log \left(1 - u1 \cdot u1\right) + \color{blue}{\left(-\mathsf{log1p}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [<=] 3.0	\[ \sqrt{-\color{blue}{\left(\log \left(1 - u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 3.0	\[ \sqrt{-\left(\log \color{blue}{\left(1 + \left(-u1 \cdot u1\right)\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   log1p-def [=>] 0.3	\[ \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(-u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   distribute-rgt-neg-in [=>] 0.3	\[ \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{u1 \cdot \left(-u1\right)}\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Taylor expanded in u2 around inf 14.4

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

5. Simplified 0.3

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

   [Start] 14.4	\[ \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]
   *-commutative [=>] 14.4	\[ \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
   log1p-def [=>] 3.0	\[ \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
   log1p-def [=>] 0.3	\[ \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(-1 \cdot {u1}^{2}\right)}} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
   mul-1-neg [=>] 0.3	\[ \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{-{u1}^{2}}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
   unpow2 [=>] 0.3	\[ \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(-\color{blue}{u1 \cdot u1}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
   distribute-rgt-neg-in [=>] 0.3	\[ \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{u1 \cdot \left(-u1\right)}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.3
Cost	16676

\[\begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999998211860657:\\ \;\;\;\;t_0 \cdot \sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	13348

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

Alternative 3
Error	3.0
Cost	13156

\[\begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.007499999832361937:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	13056

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

Alternative 5
Error	6.3
Cost	6496

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

Alternative 6
Error	7.4
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 7
Error	7.8
Cost	3552

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

Alternative 8
Error	8.7
Cost	3424

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

Alternative 9
Error	11.2
Cost	3232

\[\sqrt{u1} \]

Alternative 10
Error	25.8
Cost	32

\[u1 \]

Reproduce

herbie shell --seed 2023034 
(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))))