GTR1 distribution

Average Error: 0.5 → 0.5

Time: 15.7s

Precision: binary32

Cost: 10368

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

Math

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

↓

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

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

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

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 = log(Float32(alpha * alpha))
	return Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(Float32(pi) * Float32(t_0 + Float32(Float32(cosTheta * Float32(Float32(-1.0) + Float32(alpha * alpha))) * Float32(cosTheta * t_0)))))
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 = log((alpha * alpha));
	tmp = ((alpha * alpha) + single(-1.0)) / (single(pi) * (t_0 + ((cosTheta * (single(-1.0) + (alpha * alpha))) * (cosTheta * t_0))));
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

   \[\leadsto \color{blue}{\frac{\alpha \cdot \alpha + -1}{\pi \cdot \left(\left(1 + \left(\alpha \cdot \alpha + -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]
   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_best_oopsla_all_46_json_45_simplify-45 [=>] 0.5	\[ \frac{\color{blue}{\alpha \cdot \alpha + -1}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\color{blue}{\pi \cdot \left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\pi \cdot \left(\left(1 + \color{blue}{cosTheta \cdot \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right)}\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\pi \cdot \left(\left(1 + \color{blue}{\left(\alpha \cdot \alpha - 1\right) \cdot \left(cosTheta \cdot cosTheta\right)}\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-45 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\pi \cdot \left(\left(1 + \color{blue}{\left(\alpha \cdot \alpha + -1\right)} \cdot \left(cosTheta \cdot cosTheta\right)\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

3. Applied egg-rr 0.5

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

4. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\pi \cdot \log \left(\alpha \cdot \alpha\right) + \pi \cdot \left(\left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot cosTheta\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi} + \pi \cdot \left(\left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot cosTheta\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\color{blue}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) + \left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot cosTheta\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) + \color{blue}{\left(cosTheta \cdot \left(\alpha \cdot \alpha + -1\right)\right)} \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot cosTheta\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) + \left(cosTheta \cdot \color{blue}{\left(-1 + \alpha \cdot \alpha\right)}\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot cosTheta\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \frac{\alpha \cdot \alpha + -1}{\pi \cdot \left(\log \left(\alpha \cdot \alpha\right) + \left(cosTheta \cdot \left(-1 + \alpha \cdot \alpha\right)\right) \cdot \color{blue}{\left(cosTheta \cdot \log \left(\alpha \cdot \alpha\right)\right)}\right)} \]

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	7104

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

Alternative 2
Error	0.5
Cost	7104

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

Alternative 3
Error	0.8
Cost	6912

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

Alternative 4
Error	0.8
Cost	6912

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

Alternative 5
Error	10.6
Cost	6784

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

Alternative 6
Error	10.6
Cost	6784

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

Alternative 7
Error	1.6
Cost	6784

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

Reproduce

herbie shell --seed 2023090 
(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)))))