Average Error: 0.5 → 0.5
Time: 17.0s
Precision: binary32
Cost: 16512
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]
(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (- (* alpha alpha) 1.0)
  (*
   (* PI (log (* alpha alpha)))
   (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))
↓
(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (/ (/ (fma alpha alpha -1.0) (* (log alpha) 2.0)) PI)
  (fma (fma alpha alpha -1.0) (* cosTheta cosTheta) 1.0)))
float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((((alpha * alpha) - 1.0f) * cosTheta) * cosTheta)));
}
↓
float code(float cosTheta, float alpha) {
	return ((fmaf(alpha, alpha, -1.0f) / (logf(alpha) * 2.0f)) / ((float) M_PI)) / fmaf(fmaf(alpha, alpha, -1.0f), (cosTheta * cosTheta), 1.0f);
}

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) * cosTheta) * cosTheta))))
end
↓
function code(cosTheta, alpha)
	return Float32(Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(log(alpha) * Float32(2.0))) / Float32(pi)) / fma(fma(alpha, alpha, Float32(-1.0)), Float32(cosTheta * cosTheta), Float32(1.0)))
end

\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}
↓
\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}

Error

Derivation

  1. Initial program 0.5

    \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}} \]
    Proof

    [Start]0.5

    \[ \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

    associate-/r* [=>]0.5

    \[ \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]

    difference-of-sqr-1 [=>]0.6

    \[ \frac{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    *-commutative [=>]0.6

    \[ \frac{\frac{\color{blue}{\left(\alpha - 1\right) \cdot \left(\alpha + 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    *-lft-identity [<=]0.6

    \[ \frac{\frac{\left(\alpha - 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{1 \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    times-frac [=>]0.6

    \[ \frac{\color{blue}{\frac{\alpha - 1}{1} \cdot \frac{\alpha + 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    *-commutative [<=]0.6

    \[ \frac{\color{blue}{\frac{\alpha + 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)} \cdot \frac{\alpha - 1}{1}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    times-frac [<=]0.6

    \[ \frac{\color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    difference-of-sqr-1 [<=]0.5

    \[ \frac{\frac{\color{blue}{\alpha \cdot \alpha - 1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot 1}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    associate-/l/ [<=]0.5

    \[ \frac{\color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{1}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    *-commutative [=>]0.5

    \[ \frac{\frac{\frac{\alpha \cdot \alpha - 1}{1}}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    associate-/r* [=>]0.5

    \[ \frac{\color{blue}{\frac{\frac{\frac{\alpha \cdot \alpha - 1}{1}}{\log \left(\alpha \cdot \alpha\right)}}{\pi}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    /-rgt-identity [=>]0.5

    \[ \frac{\frac{\frac{\color{blue}{\alpha \cdot \alpha - 1}}{\log \left(\alpha \cdot \alpha\right)}}{\pi}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    fma-neg [=>]0.5

    \[ \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}}{\log \left(\alpha \cdot \alpha\right)}}{\pi}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    metadata-eval [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\log \left(\alpha \cdot \alpha\right)}}{\pi}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    log-prod [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\log \alpha + \log \alpha}}}{\pi}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    count-2 [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{2 \cdot \log \alpha}}}{\pi}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    *-commutative [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\log \alpha \cdot 2}}}{\pi}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

    +-commutative [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\color{blue}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta + 1}} \]

    associate-*l* [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\color{blue}{\left(\alpha \cdot \alpha - 1\right) \cdot \left(cosTheta \cdot cosTheta\right)} + 1} \]

    fma-def [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\color{blue}{\mathsf{fma}\left(\alpha \cdot \alpha - 1, cosTheta \cdot cosTheta, 1\right)}} \]

    fma-neg [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}, cosTheta \cdot cosTheta, 1\right)} \]

    metadata-eval [=>]0.5

    \[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right), cosTheta \cdot cosTheta, 1\right)} \]

  3. Final simplification0.5

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

Alternatives

Alternative 1
Error0.5
Cost13376
\[\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{1 + \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)} \]
Alternative 2
Error0.4
Cost10272
\[\begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right)} \end{array} \]
Alternative 3
Error0.5
Cost7104
\[\begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \end{array} \]
Alternative 4
Error0.8
Cost6912
\[\frac{-1 + \alpha \cdot \alpha}{2 \cdot \left(\pi \cdot \left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)\right)} \]
Alternative 5
Error0.8
Cost6912
\[\frac{\frac{1 - \alpha \cdot \alpha}{\pi \cdot -2}}{\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]
Alternative 6
Error0.8
Cost6912
\[\frac{-1 + \alpha \cdot \alpha}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]
Alternative 7
Error1.6
Cost6784
\[0.5 \cdot \left(\frac{\alpha + 1}{\log \alpha} \cdot \frac{\alpha + -1}{\pi}\right) \]
Alternative 8
Error1.6
Cost6784
\[0.5 \cdot \frac{\left(\alpha + -1\right) \cdot \left(\alpha + 1\right)}{\log \alpha \cdot \pi} \]
Alternative 9
Error1.6
Cost6784
\[\frac{\alpha + 1}{\pi} \cdot \frac{\alpha + -1}{\log \alpha \cdot 2} \]
Alternative 10
Error1.6
Cost6784
\[\frac{\frac{0.5 \cdot \left(\alpha + -1\right)}{\log \alpha}}{\frac{\pi}{\alpha + 1}} \]
Alternative 11
Error10.6
Cost6720
\[\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} \]
Alternative 12
Error10.6
Cost6720
\[\frac{\frac{-0.5}{\log \alpha}}{\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]
Alternative 13
Error10.6
Cost6720
\[\frac{\frac{-0.5}{\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)}}{\pi} \]
Alternative 14
Error11.0
Cost6528
\[\frac{-0.5}{\log \alpha \cdot \pi} \]
Alternative 15
Error11.0
Cost6528
\[\frac{\frac{-0.5}{\log \alpha}}{\pi} \]
Alternative 16
Error32.0
Cost3392
\[\frac{-1}{\pi \cdot \frac{0}{0}} \]

Error

Reproduce

herbie shell --seed 2022354 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0)) (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))