Beckmann Sample, near normal, slope_x

Average Error: 13.5 → 0.4

Time: 13.5s

Precision: binary32

Cost: 19456

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

Math

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

↓

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

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 (expm1 (cos (* PI (+ u2 u2)))))))

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(expm1f(cosf((((float) M_PI) * (u2 + u2)))));
}

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)))) * log1p(expm1(cos(Float32(Float32(pi) * Float32(u2 + u2))))))
end

Derivation

1. Initial program 13.5

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

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] 13.5	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   sub-neg [=>] 13.5	\[ \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.2
Cost	13476

\[\begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.0008999999845400453:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 + \left(u1 \cdot u1\right) \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)} \cdot \cos t_0\\ \end{array} \]

Alternative 2
Error	3.0
Cost	13156

\[\begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.005200000014156103:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	13056

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

Alternative 4
Error	2.1
Cost	10116

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9955000281333923:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(u1 \cdot \left(-u1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \end{array} \]

Alternative 5
Error	6.5
Cost	6496

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

Alternative 6
Error	7.7
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 7
Error	8.1
Cost	3552

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

Alternative 8
Error	8.9
Cost	3424

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

Alternative 9
Error	11.4
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023059 
(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))))