Beckmann Distribution sample, tan2theta, alphax == alphay

Average Accuracy: 55.8% → 99.0%

Time: 9.7s

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

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

↓

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

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 55.8%

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

2. Applied egg-rr 99.0%

   \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(0 + \mathsf{log1p}\left(-u0\right)\right)} \]

3. Simplified 99.0%

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

   [Start] 99.0	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(0 + \mathsf{log1p}\left(-u0\right)\right) \]
   +-lft-identity [=>] 99.0	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\mathsf{log1p}\left(-u0\right)} \]

4. Final simplification 99.0%

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

Alternatives

Alternative 1
Accuracy	99.0%
Cost	3424

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

Alternative 2
Accuracy	91.5%
Cost	864

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

Alternative 3
Accuracy	91.5%
Cost	864

\[\begin{array}{l} t_0 := u0 \cdot \left(\alpha \cdot \alpha\right)\\ t_0 + \left(\left(u0 \cdot \left(u0 \cdot 0.3333333333333333\right)\right) \cdot t_0 + t_0 \cdot \left(u0 \cdot 0.5\right)\right) \end{array} \]

Alternative 4
Accuracy	91.5%
Cost	672

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

Alternative 5
Accuracy	91.5%
Cost	480

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

Alternative 6
Accuracy	91.5%
Cost	480

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

Alternative 7
Accuracy	87.4%
Cost	352

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

Alternative 8
Accuracy	87.3%
Cost	352

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

Alternative 9
Accuracy	74.6%
Cost	160

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

Reproduce

herbie shell --seed 2023122 
(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))))