Average Error: 14.2 → 0.3
Time: 8.9s
Precision: binary32
Cost: 3424
\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]
\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
↓
\[\alpha \cdot \left(\alpha \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)\right) \]
(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))
↓
(FPCore (alpha u0) :precision binary32 (* alpha (* alpha (- (log1p (- u0))))))
float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}
↓
float code(float alpha, float u0) {
	return alpha * (alpha * -log1pf(-u0));
}

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end
↓
function code(alpha, u0)
	return Float32(alpha * Float32(alpha * Float32(-log1p(Float32(-u0)))))
end

\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)
↓
\alpha \cdot \left(\alpha \cdot \left(-\mathsf{log1p}\left(-u0\right)\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.2

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right)} \]
    Proof
    (*.f32 alpha (*.f32 (neg.f32 alpha) (log1p.f32 (neg.f32 u0)))): 0 points increase in error, 0 points decrease in error
    (*.f32 alpha (*.f32 (neg.f32 alpha) (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 u0)))))): 237 points increase in error, 0 points decrease in error
    (*.f32 alpha (*.f32 (neg.f32 alpha) (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 u0))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 alpha (neg.f32 alpha)) (log.f32 (-.f32 1 u0)))): 33 points increase in error, 31 points decrease in error
    (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (neg.f32 alpha) alpha)) (log.f32 (-.f32 1 u0))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost3424
\[\mathsf{log1p}\left(-u0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right) \]
Alternative 2
Error2.1
Cost608
\[\left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot \left(-0.5 + u0 \cdot \left(-0.3333333333333333 + u0 \cdot -0.25\right)\right)\right) \]
Alternative 3
Error2.1
Cost608
\[\alpha \cdot \left(\alpha \cdot \left(u0 - \left(u0 \cdot u0\right) \cdot \left(-0.5 + u0 \cdot \left(-0.3333333333333333 + u0 \cdot -0.25\right)\right)\right)\right) \]
Alternative 4
Error2.8
Cost480
\[u0 \cdot \left(\alpha \cdot \left(\alpha \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right) + 1\right)\right)\right) \]
Alternative 5
Error2.7
Cost480
\[\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right) \]
Alternative 6
Error2.7
Cost480
\[\alpha \cdot \left(\alpha \cdot \left(u0 + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right) \]
Alternative 7
Error4.0
Cost352
\[u0 \cdot \left(\alpha \cdot \left(\alpha + 0.5 \cdot \left(\alpha \cdot u0\right)\right)\right) \]
Alternative 8
Error4.0
Cost352
\[\alpha \cdot \left(\alpha \cdot \left(u0 + u0 \cdot \left(u0 \cdot 0.5\right)\right)\right) \]
Alternative 9
Error8.1
Cost160
\[u0 \cdot \left(\alpha \cdot \alpha\right) \]

Error

Reproduce

herbie shell --seed 2022308 
(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))))