Average Error: 0.5 → 0.5
Time: 18.4s
Precision: binary32
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\begin{array}{l} t_0 := \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)\\ \mathsf{fma}\left(\frac{1}{t_0}, \frac{\alpha \cdot \alpha}{t_1}, \frac{-1}{t_0 \cdot t_1}\right) \end{array} \]
(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
 (let* ((t_0 (log (pow (* alpha alpha) PI)))
        (t_1 (fma (fma alpha alpha -1.0) (* cosTheta cosTheta) 1.0)))
   (fma (/ 1.0 t_0) (/ (* alpha alpha) t_1) (/ -1.0 (* t_0 t_1)))))
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) {
	float t_0 = logf(powf((alpha * alpha), ((float) M_PI)));
	float t_1 = fmaf(fmaf(alpha, alpha, -1.0f), (cosTheta * cosTheta), 1.0f);
	return fmaf((1.0f / t_0), ((alpha * alpha) / t_1), (-1.0f / (t_0 * t_1)));
}

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)
	t_0 = log((Float32(alpha * alpha) ^ Float32(pi)))
	t_1 = fma(fma(alpha, alpha, Float32(-1.0)), Float32(cosTheta * cosTheta), Float32(1.0))
	return fma(Float32(Float32(1.0) / t_0), Float32(Float32(alpha * alpha) / t_1), Float32(Float32(-1.0) / Float32(t_0 * t_1)))
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)}
↓
\begin{array}{l}
t_0 := \log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)\\
t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)\\
\mathsf{fma}\left(\frac{1}{t_0}, \frac{\alpha \cdot \alpha}{t_1}, \frac{-1}{t_0 \cdot t_1}\right)
\end{array}

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

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

  3. Applied egg-rr0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2022131 
(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)))))