GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 17.4s

Alternatives: 9

Speedup: 0.7×

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: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (log (pow (* alpha alpha) PI)) (+ 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 * alpha), ((float) M_PI))) * (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((Float32(alpha * alpha) ^ Float32(pi))) * 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 * alpha) ^ single(pi))) * (single(1.0) + (cosTheta * (t_0 * cosTheta))));
end

Derivation

1. Initial program 98.7%

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

   1. add-log-exp 98.7%

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

   2. *-commutative 98.7%

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

   3. exp-to-pow 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

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(t_0 * Float32(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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in alpha around 0 98.8%

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

4. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in alpha around 0 97.1%

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

   1. associate-*r* 97.1%

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

   2. mul-1-neg 97.1%

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

   3. sub-neg 97.1%

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

   4. unpow2 97.1%

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

   5. associate-*r* 97.1%

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

   6. associate-*r* 97.1%

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

   7. associate-*r* 97.1%

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

9. Simplified 97.1%

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

10. Final simplification 97.1%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in alpha around 0 97.1%

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

   1. associate-*r* 97.1%

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

   2. mul-1-neg 97.1%

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

   3. sub-neg 97.1%

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

   4. unpow2 97.1%

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

   5. associate-*r* 97.1%

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

   6. *-commutative 97.1%

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

   7. *-commutative 97.1%

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

   8. associate-*l* 97.1%

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

   9. associate-*r* 97.1%

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

   10. *-commutative 97.1%

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

9. Simplified 97.1%

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

10. Final simplification 97.1%

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

Alternative 5: 97.7% 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 98.7%

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

2. Taylor expanded in alpha around 0 97.2%

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

4. Simplified 97.2%

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

5. Final simplification 97.2%

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

Alternative 6: 67.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 98.7%

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

6. Simplified 98.7%

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

7. Taylor expanded in alpha around 0 66.9%

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

   1. mul-1-neg 66.9%

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

   2. unsub-neg 66.9%

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

   3. unpow2 66.9%

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

9. Simplified 66.9%

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

10. Final simplification 66.9%

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

Alternative 7: 95.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in cosTheta around 0 93.5%

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

   1. *-commutative 93.5%

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

   2. associate-*l* 93.5%

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

9. Simplified 93.5%

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

10. Final simplification 93.5%

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

Alternative 8: 1.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 98.7%

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

6. Simplified 98.7%

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

7. Taylor expanded in cosTheta around inf 1.9%

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

   1. *-commutative 1.9%

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

   2. associate-/r* 1.9%

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

   3. unpow2 1.9%

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

9. Simplified 1.9%

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

10. Taylor expanded in alpha around inf 1.9%

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

    1. associate-/r* 1.9%

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

    2. log-rec 1.9%

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

12. Simplified 1.9%

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

13. Final simplification 1.9%

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

Alternative 9: 1.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. add-log-exp 98.7%

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

   2. *-commutative 98.7%

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

   3. exp-to-pow 98.8%

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

3. Applied egg-rr 98.8%

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

4. Taylor expanded in cosTheta around inf 1.9%

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

   1. *-commutative 1.9%

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

   2. associate-/r* 1.9%

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

   3. log-pow 1.9%

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

   4. unpow2 1.9%

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

   5. log-prod 1.9%

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

   6. distribute-lft-out 1.9%

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

   7. count-2 1.9%

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

   8. associate-/r* 1.9%

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

   9. metadata-eval 1.9%

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

   10. associate-/r* 1.9%

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

   11. associate-*l* 1.9%

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

   12. unpow2 1.9%

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

6. Simplified 1.9%

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

7. Final simplification 1.9%

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

Reproduce

herbie shell --seed 2023279 
(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)))))