GTR1 distribution

Average Error: 0.5 → 0.4

Time: 15.9s

Precision: binary32

Cost: 10272

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{\frac{t_0}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}}{1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)} \end{array} \]

FPCore

(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 (+ (* alpha alpha) -1.0)))
   (/
    (/ t_0 (log (pow (* alpha alpha) PI)))
    (+ 1.0 (* cosTheta (* t_0 cosTheta))))))

C

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 = (alpha * alpha) + -1.0f;
	return (t_0 / logf(powf((alpha * alpha), ((float) M_PI)))) / (1.0f + (cosTheta * (t_0 * cosTheta)));
}

Julia

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 = Float32(Float32(alpha * alpha) + Float32(-1.0))
	return Float32(Float32(t_0 / log((Float32(alpha * alpha) ^ Float32(pi)))) / Float32(Float32(1.0) + Float32(cosTheta * Float32(t_0 * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) + ((((alpha * alpha) - single(1.0)) * cosTheta) * cosTheta)));
end

↓

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) + single(-1.0);
	tmp = (t_0 / log(((alpha * alpha) ^ single(pi)))) / (single(1.0) + (cosTheta * (t_0 * cosTheta)));
end

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. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
   rational.json-simplify-46 [=>] 0.5	\[ \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-17 [=>] 0.5	\[ \frac{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{\color{blue}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1}} \]
   rational.json-simplify-50 [=>] 0.5	\[ \color{blue}{\frac{-\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{-1 - \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-50 [=>] 0.5	\[ \color{blue}{\frac{-\left(-\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}\right)}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1}} \]
   rational.json-simplify-12 [=>] 0.5	\[ \frac{\color{blue}{0 - \left(-\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}\right)}}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1} \]
   rational.json-simplify-12 [=>] 0.5	\[ \frac{0 - \color{blue}{\left(0 - \frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}\right)}}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1} \]
   rational.json-simplify-45 [=>] 0.5	\[ \frac{\color{blue}{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)} - \left(0 - 0\right)}}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1} \]
   metadata-eval [=>] 0.5	\[ \frac{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)} - \color{blue}{0}}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1} \]
   rational.json-simplify-5 [=>] 0.5	\[ \frac{\color{blue}{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1} \]
   rational.json-simplify-16 [=>] 0.5	\[ \frac{\frac{\color{blue}{\alpha \cdot \alpha + -1}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta - -1} \]
   rational.json-simplify-17 [<=] 0.5	\[ \frac{\frac{\alpha \cdot \alpha + -1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{\color{blue}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{\frac{\alpha \cdot \alpha + -1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \color{blue}{cosTheta \cdot \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right)}} \]

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.6
Cost	7104

\[\frac{\left(1 - \alpha \cdot \alpha\right) \cdot \frac{\frac{-0.5}{\log \alpha}}{\pi}}{1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right)} \]

Alternative 2
Error	0.5
Cost	7104

\[\begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t_0}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(t_0 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \]

Alternative 3
Error	0.5
Cost	7104

\[\begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{\frac{t_0}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)} \end{array} \]

Alternative 4
Error	0.8
Cost	6944

\[\frac{\frac{0.5}{\log \alpha \cdot \pi} \cdot \left(\alpha \cdot \alpha + -1\right)}{1 + cosTheta \cdot \left(-cosTheta\right)} \]

Alternative 5
Error	0.8
Cost	6944

\[\frac{\frac{\alpha \cdot \alpha + -1}{2 \cdot \left(\log \alpha \cdot \pi\right)}}{1 + cosTheta \cdot \left(-cosTheta\right)} \]

Alternative 6
Error	0.8
Cost	6944

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

Alternative 7
Error	10.5
Cost	6880

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

Alternative 8
Error	10.5
Cost	6816

\[\frac{\frac{\frac{0.5}{\pi}}{\log \left(\frac{1}{\alpha}\right)}}{1 + cosTheta \cdot \left(-cosTheta\right)} \]

Alternative 9
Error	10.5
Cost	6784

\[\frac{1}{\pi} \cdot \frac{-0.5}{\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]

Alternative 10
Error	10.5
Cost	6752

\[\frac{\frac{-0.5}{\log \alpha \cdot \pi}}{1 + cosTheta \cdot \left(-cosTheta\right)} \]

Alternative 11
Error	10.5
Cost	6720

\[\frac{-0.5}{\left(\log \alpha \cdot \pi\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]

Alternative 12
Error	10.5
Cost	6720

\[\frac{\frac{\frac{-0.5}{\log \alpha}}{\pi}}{1 - cosTheta \cdot cosTheta} \]

Alternative 13
Error	10.9
Cost	6528

\[\frac{-0.5}{\log \alpha \cdot \pi} \]

Alternative 14
Error	10.9
Cost	6528

\[\frac{\frac{-0.5}{\pi}}{\log \alpha} \]

Reproduce

herbie shell --seed 2023068 
(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)))))