GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 12.1s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

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

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 0.7× speedup

Math

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

FPCore

(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) {
	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)
	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)
	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 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. pow2 98.6%

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

   2. log-pow 98.5%

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

   3. associate-*l* 98.5%

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

   4. add-log-exp 98.5%

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

   5. *-commutative 98.5%

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

   6. exp-to-pow 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (+ 1.0 (* cosTheta (* t_0 cosTheta))) (* 2.0 (* PI (log alpha)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) + -1.0f;
	return t_0 / ((1.0f + (cosTheta * (t_0 * cosTheta))) * (2.0f * (((float) M_PI) * logf(alpha))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) + Float32(-1.0))
	return Float32(t_0 / Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(t_0 * cosTheta))) * Float32(Float32(2.0) * Float32(Float32(pi) * log(alpha)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) + single(-1.0);
	tmp = t_0 / ((single(1.0) + (cosTheta * (t_0 * cosTheta))) * (single(2.0) * (single(pi) * log(alpha))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 98.5%

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

3. Final simplification 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (+ 1.0 (* cosTheta (* t_0 cosTheta))) (* PI (log (* alpha alpha)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Final simplification 98.6%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (+ (* alpha alpha) -1.0)
  (* (* PI (log (* alpha alpha))) (- 1.0 (* cosTheta cosTheta)))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f - (cosTheta * 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(cosTheta * cosTheta))))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 97.1%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
3. Step-by-step derivation

   1. mul-1-neg 97.3%

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

4. Simplified 97.1%

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

5. Final simplification 97.1%

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

Alternative 5: 95.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (+ (* alpha alpha) -1.0) (* PI (* 2.0 (log alpha)))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / (((float) M_PI) * (2.0f * logf(alpha)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) + Float32(-1.0)) / Float32(Float32(pi) * Float32(Float32(2.0) * log(alpha))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / (single(pi) * (single(2.0) * log(alpha)));
end

Derivation

1. Initial program 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Step-by-step derivation

   1. pow2 98.6%

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

   2. log-pow 98.5%

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

   3. associate-*l* 98.5%

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

   4. add-log-exp 98.5%

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

   5. *-commutative 98.5%

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

   6. exp-to-pow 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 97.3%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\alpha}^{\left(\pi \cdot 2\right)}\right) \cdot \left(1 + \color{blue}{\left(-1 \cdot cosTheta\right)} \cdot cosTheta\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in cosTheta around 0 94.2%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{2 \cdot \left(\pi \cdot \log \alpha\right)}} \]
8. Step-by-step derivation

   1. associate-*r* 94.2%

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

   2. *-commutative 94.2%

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

   3. associate-*l* 94.2%

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

9. Simplified 94.2%

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

10. Final simplification 94.2%

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

Alternative 6: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \alpha \cdot 0.3333333333333333\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -0.5 (* PI (* 3.0 (* (log alpha) 0.3333333333333333)))))

C

float code(float cosTheta, float alpha) {
	return -0.5f / (((float) M_PI) * (3.0f * (logf(alpha) * 0.3333333333333333f)));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(-0.5) / Float32(Float32(pi) * Float32(Float32(3.0) * Float32(log(alpha) * Float32(0.3333333333333333)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-0.5) / (single(pi) * (single(3.0) * (log(alpha) * single(0.3333333333333333))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 66.7%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. associate-*r* 66.7%

      \[\leadsto \frac{-0.5}{\color{blue}{\left(\pi \cdot \log \alpha\right) \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)}} \]

   2. mul-1-neg 66.7%

      \[\leadsto \frac{-0.5}{\left(\pi \cdot \log \alpha\right) \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)} \]

   3. unsub-neg 66.7%

      \[\leadsto \frac{-0.5}{\left(\pi \cdot \log \alpha\right) \cdot \color{blue}{\left(1 - {cosTheta}^{2}\right)}} \]

4. Simplified 66.7%

   \[\leadsto \color{blue}{\frac{-0.5}{\left(\pi \cdot \log \alpha\right) \cdot \left(1 - {cosTheta}^{2}\right)}} \]

5. Taylor expanded in cosTheta around 0 65.0%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \log \alpha}} \]
6. Step-by-step derivation

   1. add-cube-cbrt 65.0%

      \[\leadsto \frac{-0.5}{\pi \cdot \log \color{blue}{\left(\left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) \cdot \sqrt[3]{\alpha}\right)}} \]

   2. log-prod 65.0%

      \[\leadsto \frac{-0.5}{\pi \cdot \color{blue}{\left(\log \left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) + \log \left(\sqrt[3]{\alpha}\right)\right)}} \]

   3. pow2 65.0%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{\alpha}\right)}^{2}\right)} + \log \left(\sqrt[3]{\alpha}\right)\right)} \]

7. Applied egg-rr 65.0%

   \[\leadsto \frac{-0.5}{\pi \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{\alpha}\right)}^{2}\right) + \log \left(\sqrt[3]{\alpha}\right)\right)}} \]
8. Step-by-step derivation

   1. log-pow 65.0%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\alpha}\right)} + \log \left(\sqrt[3]{\alpha}\right)\right)} \]

   2. distribute-lft1-in 65.0%

      \[\leadsto \frac{-0.5}{\pi \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\alpha}\right)\right)}} \]

   3. metadata-eval 65.0%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{\alpha}\right)\right)} \]

9. Simplified 65.0%

   \[\leadsto \frac{-0.5}{\pi \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\alpha}\right)\right)}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 65.0%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \log \color{blue}{\left(\sqrt{\sqrt[3]{\alpha}} \cdot \sqrt{\sqrt[3]{\alpha}}\right)}\right)} \]

    2. log-prod 65.0%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \color{blue}{\left(\log \left(\sqrt{\sqrt[3]{\alpha}}\right) + \log \left(\sqrt{\sqrt[3]{\alpha}}\right)\right)}\right)} \]

    3. pow1/3 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \left(\sqrt{\color{blue}{{\alpha}^{0.3333333333333333}}}\right) + \log \left(\sqrt{\sqrt[3]{\alpha}}\right)\right)\right)} \]

    4. sqrt-pow1 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \color{blue}{\left({\alpha}^{\left(\frac{0.3333333333333333}{2}\right)}\right)} + \log \left(\sqrt{\sqrt[3]{\alpha}}\right)\right)\right)} \]

    5. metadata-eval 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \left({\alpha}^{\color{blue}{0.16666666666666666}}\right) + \log \left(\sqrt{\sqrt[3]{\alpha}}\right)\right)\right)} \]

    6. pow1/3 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \left({\alpha}^{0.16666666666666666}\right) + \log \left(\sqrt{\color{blue}{{\alpha}^{0.3333333333333333}}}\right)\right)\right)} \]

    7. sqrt-pow1 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \left({\alpha}^{0.16666666666666666}\right) + \log \color{blue}{\left({\alpha}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}\right)\right)} \]

    8. metadata-eval 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \left({\alpha}^{0.16666666666666666}\right) + \log \left({\alpha}^{\color{blue}{0.16666666666666666}}\right)\right)\right)} \]

11. Applied egg-rr 65.1%

    \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \color{blue}{\left(\log \left({\alpha}^{0.16666666666666666}\right) + \log \left({\alpha}^{0.16666666666666666}\right)\right)}\right)} \]
12. Step-by-step derivation

    1. log-pow 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\color{blue}{0.16666666666666666 \cdot \log \alpha} + \log \left({\alpha}^{0.16666666666666666}\right)\right)\right)} \]

    2. log-pow 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(0.16666666666666666 \cdot \log \alpha + \color{blue}{0.16666666666666666 \cdot \log \alpha}\right)\right)} \]

    3. distribute-rgt-out 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \color{blue}{\left(\log \alpha \cdot \left(0.16666666666666666 + 0.16666666666666666\right)\right)}\right)} \]

    4. metadata-eval 65.1%

       \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \alpha \cdot \color{blue}{0.3333333333333333}\right)\right)} \]

13. Simplified 65.1%

    \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \color{blue}{\left(\log \alpha \cdot 0.3333333333333333\right)}\right)} \]

14. Final simplification 65.1%

    \[\leadsto \frac{-0.5}{\pi \cdot \left(3 \cdot \left(\log \alpha \cdot 0.3333333333333333\right)\right)} \]

Alternative 7: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\pi} \cdot \frac{-1}{\log \alpha} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (* (/ 0.5 PI) (/ -1.0 (log alpha))))

C

float code(float cosTheta, float alpha) {
	return (0.5f / ((float) M_PI)) * (-1.0f / logf(alpha));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(0.5) / Float32(pi)) * Float32(Float32(-1.0) / log(alpha)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(0.5) / single(pi)) * (single(-1.0) / log(alpha));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in cosTheta around 0 94.3%

   \[\leadsto \color{blue}{\frac{{\alpha}^{2} - 1}{\pi \cdot \log \left({\alpha}^{2}\right)}} \]
3. Step-by-step derivation

   1. associate-/r* 94.1%

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

   2. unpow2 94.1%

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

   3. fma-neg 94.0%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}}{\pi}}{\log \left({\alpha}^{2}\right)} \]

   4. metadata-eval 94.0%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\pi}}{\log \left({\alpha}^{2}\right)} \]

   5. *-lft-identity 94.0%

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}}{\log \left({\alpha}^{2}\right)} \]

   6. log-pow 94.1%

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

   7. times-frac 94.1%

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

   8. metadata-eval 94.1%

      \[\leadsto \color{blue}{0.5} \cdot \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}{\log \alpha} \]

   9. metadata-eval 94.1%

      \[\leadsto 0.5 \cdot \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\pi}}{\log \alpha} \]

   10. fma-neg 94.1%

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

   11. unpow2 94.1%

       \[\leadsto 0.5 \cdot \frac{\frac{\color{blue}{{\alpha}^{2}} - 1}{\pi}}{\log \alpha} \]

   12. associate-/r* 94.2%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{{\alpha}^{2} - 1}{\pi \cdot \log \alpha}} \]

   13. associate-*r/ 94.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left({\alpha}^{2} - 1\right)}{\pi \cdot \log \alpha}} \]

   14. unpow2 94.2%

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

   15. fma-neg 94.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}}{\pi \cdot \log \alpha} \]

   16. metadata-eval 94.2%

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\pi \cdot \log \alpha} \]

   17. times-frac 93.9%

       \[\leadsto \color{blue}{\frac{0.5}{\pi} \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}} \]

4. Simplified 93.9%

   \[\leadsto \color{blue}{\frac{0.5}{\pi} \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}} \]

5. Taylor expanded in alpha around 0 65.1%

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

6. Final simplification 65.1%

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

Alternative 8: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\pi \cdot \log \alpha} \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ -0.5 (* PI (log alpha))))

C

float code(float cosTheta, float alpha) {
	return -0.5f / (((float) M_PI) * logf(alpha));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(-0.5) / Float32(Float32(pi) * log(alpha)))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-0.5) / (single(pi) * log(alpha));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 66.7%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. associate-*r* 66.7%

      \[\leadsto \frac{-0.5}{\color{blue}{\left(\pi \cdot \log \alpha\right) \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)}} \]

   2. mul-1-neg 66.7%

      \[\leadsto \frac{-0.5}{\left(\pi \cdot \log \alpha\right) \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)} \]

   3. unsub-neg 66.7%

      \[\leadsto \frac{-0.5}{\left(\pi \cdot \log \alpha\right) \cdot \color{blue}{\left(1 - {cosTheta}^{2}\right)}} \]

4. Simplified 66.7%

   \[\leadsto \color{blue}{\frac{-0.5}{\left(\pi \cdot \log \alpha\right) \cdot \left(1 - {cosTheta}^{2}\right)}} \]

5. Taylor expanded in cosTheta around 0 65.0%

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

6. Final simplification 65.0%

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

Alternative 9: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5}{\pi}}{\log \alpha} \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ (/ -0.5 PI) (log alpha)))

C

float code(float cosTheta, float alpha) {
	return (-0.5f / ((float) M_PI)) / logf(alpha);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(-0.5) / Float32(pi)) / log(alpha))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-0.5) / single(pi)) / log(alpha);
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in cosTheta around 0 94.3%

   \[\leadsto \color{blue}{\frac{{\alpha}^{2} - 1}{\pi \cdot \log \left({\alpha}^{2}\right)}} \]
3. Step-by-step derivation

   1. associate-/r* 94.1%

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

   2. unpow2 94.1%

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

   3. fma-neg 94.0%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}}{\pi}}{\log \left({\alpha}^{2}\right)} \]

   4. metadata-eval 94.0%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\pi}}{\log \left({\alpha}^{2}\right)} \]

   5. *-lft-identity 94.0%

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}}{\log \left({\alpha}^{2}\right)} \]

   6. log-pow 94.1%

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

   7. times-frac 94.1%

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

   8. metadata-eval 94.1%

      \[\leadsto \color{blue}{0.5} \cdot \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\pi}}{\log \alpha} \]

   9. metadata-eval 94.1%

      \[\leadsto 0.5 \cdot \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\pi}}{\log \alpha} \]

   10. fma-neg 94.1%

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

   11. unpow2 94.1%

       \[\leadsto 0.5 \cdot \frac{\frac{\color{blue}{{\alpha}^{2}} - 1}{\pi}}{\log \alpha} \]

   12. associate-/r* 94.2%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{{\alpha}^{2} - 1}{\pi \cdot \log \alpha}} \]

   13. associate-*r/ 94.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left({\alpha}^{2} - 1\right)}{\pi \cdot \log \alpha}} \]

   14. unpow2 94.2%

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

   15. fma-neg 94.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}}{\pi \cdot \log \alpha} \]

   16. metadata-eval 94.2%

       \[\leadsto \frac{0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\pi \cdot \log \alpha} \]

   17. times-frac 93.9%

       \[\leadsto \color{blue}{\frac{0.5}{\pi} \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}} \]

4. Simplified 93.9%

   \[\leadsto \color{blue}{\frac{0.5}{\pi} \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}} \]

5. Taylor expanded in alpha around 0 65.1%

   \[\leadsto \frac{0.5}{\pi} \cdot \color{blue}{\frac{-1}{\log \alpha}} \]
6. Step-by-step derivation

   1. *-commutative 65.1%

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

   2. frac-times 65.0%

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

   3. metadata-eval 65.0%

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

   4. add-cube-cbrt 65.0%

      \[\leadsto \frac{-0.5}{\log \color{blue}{\left(\left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) \cdot \sqrt[3]{\alpha}\right)} \cdot \pi} \]

   5. pow3 65.0%

      \[\leadsto \frac{-0.5}{\log \color{blue}{\left({\left(\sqrt[3]{\alpha}\right)}^{3}\right)} \cdot \pi} \]

   6. exp-to-pow 65.0%

      \[\leadsto \frac{-0.5}{\log \color{blue}{\left(e^{\log \left(\sqrt[3]{\alpha}\right) \cdot 3}\right)} \cdot \pi} \]

   7. add-log-exp 65.0%

      \[\leadsto \frac{-0.5}{\color{blue}{\left(\log \left(\sqrt[3]{\alpha}\right) \cdot 3\right)} \cdot \pi} \]

   8. *-commutative 65.0%

      \[\leadsto \frac{-0.5}{\color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\alpha}\right)\right)} \cdot \pi} \]

   9. *-commutative 65.0%

      \[\leadsto \frac{-0.5}{\color{blue}{\pi \cdot \left(3 \cdot \log \left(\sqrt[3]{\alpha}\right)\right)}} \]

   10. associate-/r* 65.0%

       \[\leadsto \color{blue}{\frac{\frac{-0.5}{\pi}}{3 \cdot \log \left(\sqrt[3]{\alpha}\right)}} \]

   11. *-commutative 65.0%

       \[\leadsto \frac{\frac{-0.5}{\pi}}{\color{blue}{\log \left(\sqrt[3]{\alpha}\right) \cdot 3}} \]

   12. add-log-exp 65.0%

       \[\leadsto \frac{\frac{-0.5}{\pi}}{\color{blue}{\log \left(e^{\log \left(\sqrt[3]{\alpha}\right) \cdot 3}\right)}} \]

   13. exp-to-pow 65.0%

       \[\leadsto \frac{\frac{-0.5}{\pi}}{\log \color{blue}{\left({\left(\sqrt[3]{\alpha}\right)}^{3}\right)}} \]

   14. pow3 65.1%

       \[\leadsto \frac{\frac{-0.5}{\pi}}{\log \color{blue}{\left(\left(\sqrt[3]{\alpha} \cdot \sqrt[3]{\alpha}\right) \cdot \sqrt[3]{\alpha}\right)}} \]

   15. add-cube-cbrt 65.0%

       \[\leadsto \frac{\frac{-0.5}{\pi}}{\log \color{blue}{\alpha}} \]

7. Applied egg-rr 65.0%

   \[\leadsto \color{blue}{\frac{\frac{-0.5}{\pi}}{\log \alpha}} \]

8. Final simplification 65.0%

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

Reproduce

herbie shell --seed 2023313 
(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)))))