Beckmann Distribution sample, tan2theta, alphax == alphay

Average Error: 14.0 → 0.3

Time: 9.4s

Precision: binary32

Cost: 3424

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

↓

\[\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

↓

(FPCore (alpha u0) :precision binary32 (* (- alpha) (* alpha (log1p (- u0)))))

C

float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}

↓

float code(float alpha, float u0) {
	return -alpha * (alpha * log1pf(-u0));
}

Julia

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end

↓

function code(alpha, u0)
	return Float32(Float32(-alpha) * Float32(alpha * log1p(Float32(-u0))))
end

Derivation

1. Initial program 14.0

   \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
   Proof

   [Start] 14.0	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
   associate-*l* [=>] 14.0	\[ \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]
   sub-neg [=>] 14.0	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \log \color{blue}{\left(1 + \left(-u0\right)\right)}\right) \]
   log1p-def [=>] 0.3	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)}\right) \]

3. Final simplification 0.3

   \[\leadsto \left(-\alpha\right) \cdot \left(\alpha \cdot \mathsf{log1p}\left(-u0\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	3424

\[\mathsf{log1p}\left(-u0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right) \]

Alternative 2
Error	2.7
Cost	736

\[\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \frac{\left(u0 \cdot u0\right) \cdot \left(\left(u0 \cdot u0\right) \cdot 0.1111111111111111 + -0.25\right)}{u0 \cdot 0.3333333333333333 + -0.5}\right) \]

Alternative 3
Error	2.7
Cost	736

\[\left(\alpha \cdot \alpha\right) \cdot \left(u0 \cdot \left(u0 \cdot \left(u0 \cdot 0.3333333333333333\right)\right) + u0 \cdot \left(u0 \cdot 0.5\right)\right) + \alpha \cdot \left(\alpha \cdot u0\right) \]

Alternative 4
Error	2.4
Cost	736

\[\frac{1}{\left(\frac{\frac{1}{u0}}{\alpha \cdot \alpha} + -0.08333333333333333 \cdot \frac{u0}{\alpha \cdot \alpha}\right) + \frac{-0.5}{\alpha \cdot \alpha}} \]

Alternative 5
Error	2.7
Cost	480

\[\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right) \]

Alternative 6
Error	4.2
Cost	352

\[\alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(1 + u0 \cdot 0.5\right)\right)\right) \]

Alternative 7
Error	4.1
Cost	352

\[\alpha \cdot \left(\alpha \cdot \left(u0 + u0 \cdot \left(u0 \cdot 0.5\right)\right)\right) \]

Alternative 8
Error	4.1
Cost	352

\[\left(\alpha \cdot u0\right) \cdot \left(\alpha + \alpha \cdot \left(u0 \cdot 0.5\right)\right) \]

Alternative 9
Error	8.2
Cost	160

\[\alpha \cdot \left(\alpha \cdot u0\right) \]

Reproduce

herbie shell --seed 2023054 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))