Average Error: 0.5 → 0.4
Time: 19.1s
Precision: binary32
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\sqrt[3]{{\left(\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\alpha}^{\pi} \cdot {\alpha}^{\pi}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}\right)}^{3}} \]
(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
 (cbrt
  (pow
   (/
    (/ (fma alpha alpha -1.0) (log (* (pow alpha PI) (pow alpha PI))))
    (fma (fma alpha alpha -1.0) (* cosTheta cosTheta) 1.0))
   3.0)))
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 cbrtf(powf(((fmaf(alpha, alpha, -1.0f) / logf((powf(alpha, ((float) M_PI)) * powf(alpha, ((float) M_PI))))) / fmaf(fmaf(alpha, alpha, -1.0f), (cosTheta * cosTheta), 1.0f)), 3.0f));
}

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 cbrt((Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / log(Float32((alpha ^ Float32(pi)) * (alpha ^ Float32(pi))))) / fma(fma(alpha, alpha, Float32(-1.0)), Float32(cosTheta * cosTheta), Float32(1.0))) ^ Float32(3.0)))
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)}
↓
\sqrt[3]{{\left(\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\alpha}^{\pi} \cdot {\alpha}^{\pi}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}\right)}^{3}}

Error

Bits error versus cosTheta

Bits error versus alpha

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. Applied egg-rr0.4

    \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}\right)}^{3}}} \]

  3. Applied egg-rr0.4

    \[\leadsto \sqrt[3]{{\left(\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \color{blue}{\left({\alpha}^{\pi} \cdot {\alpha}^{\pi}\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}\right)}^{3}} \]

  4. Final simplification0.4

    \[\leadsto \sqrt[3]{{\left(\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\alpha}^{\pi} \cdot {\alpha}^{\pi}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}\right)}^{3}} \]

Reproduce

herbie shell --seed 2022153 
(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)))))