GTR1 distribution

Average Error: 0.5 → 0.4

Time: 16.5s

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

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

↓

\[\begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t_0}{\log \left({\alpha}^{\left(\pi \cdot 2\right)}\right) \cdot \left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right)} \end{array} \]

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 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (log (pow alpha (* PI 2.0))) (+ 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, (((float) M_PI) * 2.0f))) * (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(t_0 / Float32(log((alpha ^ Float32(Float32(pi) * Float32(2.0)))) * 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 ^ (single(pi) * single(2.0)))) * (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. Applied egg-rr 0.4

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

3. Taylor expanded in alpha around 0 0.5

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left(e^{2 \cdot \left(\log \alpha \cdot \pi\right)}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Simplified 0.4

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\alpha}^{\left(\pi \cdot 2\right)}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
   Proof
   (pow.f32 alpha (*.f32 (PI.f32) 2)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= exp-to-pow_binary32 (exp.f32 (*.f32 (log.f32 alpha) (*.f32 (PI.f32) 2)))): 116 points increase in error, 127 points decrease in error
   (exp.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (log.f32 alpha) (PI.f32)) 2))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (Rewrite<= *-commutative_binary32 (*.f32 2 (*.f32 (log.f32 alpha) (PI.f32))))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	10240

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

Alternative 2
Error	0.5
Cost	7104

\[\begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right) \cdot \left(\left(\pi \cdot 2\right) \cdot \log \alpha\right)} \end{array} \]

Alternative 3
Error	0.8
Cost	6912

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

Alternative 4
Error	1.6
Cost	6784

\[\frac{\frac{\alpha + 1}{\pi}}{2 \cdot \frac{\log \alpha}{\alpha + -1}} \]

Alternative 5
Error	1.6
Cost	6784

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

Alternative 6
Error	1.6
Cost	6784

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

Alternative 7
Error	10.5
Cost	6720

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

Alternative 8
Error	10.9
Cost	6592

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

Alternative 9
Error	10.9
Cost	6528

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

Reproduce

herbie shell --seed 2022310 
(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)))))