GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 21.0s

Alternatives: 11

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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 98.5%

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

   1. add-log-exp 98.5%

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

   2. *-commutative 98.5%

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

   3. exp-to-pow 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

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 98.5%

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

2. Final simplification 98.5%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. pow2 98.5%

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

   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. *-commutative 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in alpha around 0 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in alpha around 0 97.3%

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

   1. *-commutative 97.3%

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

   2. associate-*l* 97.3%

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

   3. mul-1-neg 97.3%

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

   4. unsub-neg 97.3%

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

   5. unpow2 97.3%

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

   6. *-commutative 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. pow2 98.5%

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

   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. *-commutative 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in alpha around 0 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

   2. unpow2 97.3%

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

   3. distribute-rgt-neg-out 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 5: 67.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. pow2 98.5%

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

   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. *-commutative 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in alpha around 0 66.8%

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

   1. associate-*r* 66.8%

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

   2. mul-1-neg 66.8%

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

   3. unsub-neg 66.8%

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

   4. unpow2 66.8%

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

6. Simplified 66.8%

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

7. Final simplification 66.8%

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

Alternative 6: 67.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. difference-of-sqr-1 98.1%

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

   2. *-commutative 98.1%

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

   3. +-commutative 98.1%

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

   4. fma-neg 98.1%

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

   5. metadata-eval 98.1%

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

   6. associate-*r* 98.1%

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

   7. fma-udef 98.1%

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

   8. times-frac 98.2%

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

   9. sub-neg 98.2%

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

   10. metadata-eval 98.2%

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

   11. pow2 98.2%

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

   12. log-pow 98.3%

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

   13. *-commutative 98.3%

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

3. Applied egg-rr 98.3%

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

4. Taylor expanded in alpha around 0 66.8%

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

   1. *-commutative 66.8%

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

   2. associate-*l* 66.8%

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

   3. mul-1-neg 66.8%

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

   4. unsub-neg 66.8%

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

   5. unpow2 66.8%

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

   6. *-commutative 66.8%

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

6. Simplified 66.8%

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

7. Final simplification 66.8%

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

Alternative 7: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. pow2 98.5%

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

   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. *-commutative 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in alpha around 0 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in cosTheta around 0 94.7%

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

   1. associate-*r* 94.7%

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

   2. *-commutative 94.7%

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

   3. associate-*l* 94.7%

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

9. Simplified 94.7%

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

10. Final simplification 94.7%

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

Alternative 8: 3.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0 94.7%

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

4. Simplified 94.7%

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

5. Taylor expanded in alpha around inf 3.6%

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

   1. unpow2 3.6%

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

   2. times-frac 3.6%

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

   3. log-rec 3.6%

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

7. Simplified 3.6%

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

8. Final simplification 3.6%

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

Alternative 9: 3.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0 94.7%

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

4. Simplified 94.7%

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

5. Taylor expanded in alpha around inf 3.6%

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

   1. unpow2 3.6%

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

   2. *-commutative 3.6%

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

   3. log-rec 3.6%

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

7. Simplified 3.6%

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

8. Final simplification 3.6%

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

Alternative 10: 1.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. pow2 98.5%

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

   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. *-commutative 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in alpha around inf 1.9%

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

   1. log-rec 1.9%

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

   2. unpow2 1.9%

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

6. Simplified 1.9%

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

7. Taylor expanded in alpha around 0 1.9%

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

   1. unpow2 1.9%

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

   2. associate-*r* 1.9%

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

   3. *-commutative 1.9%

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

   4. associate-*r* 1.9%

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

   5. unpow2 1.9%

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

   6. associate-*l* 1.9%

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

   7. unpow2 1.9%

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

   8. *-commutative 1.9%

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

9. Simplified 1.9%

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

10. Final simplification 1.9%

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

Alternative 11: -0.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right) \cdot \left(\pi \cdot \frac{0}{0}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/ t_0 (* (+ 1.0 (* cosTheta (* t_0 cosTheta))) (* PI (/ 0.0 0.0))))))

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) * (0.0f / 0.0f)));
}

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) * Float32(Float32(0.0) / Float32(0.0)))))
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) * (single(0.0) / single(0.0))));
end

Derivation

1. Initial program 98.5%

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

   1. log-prod 98.5%

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

   2. flip-+ -0.0%

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

   3. pow2 0.0%

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

   4. pow2 -0.0%

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

3. Applied egg-rr -0.0%

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

5. Simplified -0.0%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \color{blue}{\frac{0}{0}}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

6. Final simplification -0.0%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right)\right) \cdot \left(\pi \cdot \frac{0}{0}\right)} \]

Reproduce

herbie shell --seed 2023187 
(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)))))