GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 16.0s

Alternatives: 10

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 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. 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: 98.4% 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 98.5%

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

2. Taylor expanded in 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 98.5%

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

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 4: 97.4% 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 98.5%

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

2. Taylor expanded in alpha around 0 97.4%

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

   1. mul-1-neg 97.4%

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

4. Simplified 97.4%

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

5. Final simplification 97.4%

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

Alternative 5: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * (((alpha + single(1.0)) / single(pi)) * ((alpha + single(-1.0)) / 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. associate-/r* 98.5%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. log-prod 98.6%

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

   5. distribute-rgt-in 98.6%

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

   6. distribute-lft-out 98.6%

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

   7. *-rgt-identity 98.6%

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

   8. *-rgt-identity 98.6%

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

   9. distribute-lft-out 98.6%

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

   10. metadata-eval 98.6%

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

   11. +-commutative 98.6%

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

   12. associate-*l* 98.6%

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

   13. fma-def 98.6%

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

   14. fma-neg 98.6%

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

   15. metadata-eval 98.6%

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

3. Simplified 98.6%

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

   1. metadata-eval 98.6%

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

   2. fma-neg 98.5%

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

   3. difference-of-sqr-1 98.2%

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

   4. add-exp-log 98.2%

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

   5. expm1-udef 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \alpha\right)}}{\log \alpha \cdot \left(\pi \cdot 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\left(\pi \cdot 2\right) \cdot \log \alpha}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   7. associate-*l* 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\pi \cdot \left(2 \cdot \log \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   8. log-pow 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \color{blue}{\log \left({\alpha}^{2}\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   9. pow2 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \log \color{blue}{\left(\alpha \cdot \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   10. *-commutative 98.2%

       \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   11. times-frac 98.0%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + 1}{\log \left(\alpha \cdot \alpha\right)} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   12. log-prod 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha + \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   13. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{1 \cdot \log \alpha} + \log \alpha} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   14. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{1 \cdot \log \alpha + \color{blue}{1 \cdot \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   15. distribute-rgt-out 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha \cdot \left(1 + 1\right)}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   16. metadata-eval 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\log \alpha \cdot \color{blue}{2}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   17. expm1-udef 98.0%

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

   18. add-exp-log 97.9%

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

   19. sub-neg 97.9%

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

   20. metadata-eval 97.9%

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

5. Applied egg-rr 97.9%

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

6. Taylor expanded in cosTheta around 0 94.2%

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

   1. times-frac 94.1%

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

   2. +-commutative 94.1%

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

   3. sub-neg 94.1%

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

   4. metadata-eval 94.1%

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

8. Applied egg-rr 94.1%

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

9. Final simplification 94.1%

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

Alternative 6: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * (((alpha + single(1.0)) * (alpha + single(-1.0))) / (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. associate-/r* 98.5%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. log-prod 98.6%

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

   5. distribute-rgt-in 98.6%

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

   6. distribute-lft-out 98.6%

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

   7. *-rgt-identity 98.6%

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

   8. *-rgt-identity 98.6%

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

   9. distribute-lft-out 98.6%

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

   10. metadata-eval 98.6%

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

   11. +-commutative 98.6%

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

   12. associate-*l* 98.6%

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

   13. fma-def 98.6%

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

   14. fma-neg 98.6%

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

   15. metadata-eval 98.6%

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

3. Simplified 98.6%

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

   1. metadata-eval 98.6%

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

   2. fma-neg 98.5%

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

   3. difference-of-sqr-1 98.2%

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

   4. add-exp-log 98.2%

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

   5. expm1-udef 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \alpha\right)}}{\log \alpha \cdot \left(\pi \cdot 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\left(\pi \cdot 2\right) \cdot \log \alpha}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   7. associate-*l* 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\pi \cdot \left(2 \cdot \log \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   8. log-pow 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \color{blue}{\log \left({\alpha}^{2}\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   9. pow2 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \log \color{blue}{\left(\alpha \cdot \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   10. *-commutative 98.2%

       \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   11. times-frac 98.0%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + 1}{\log \left(\alpha \cdot \alpha\right)} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   12. log-prod 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha + \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   13. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{1 \cdot \log \alpha} + \log \alpha} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   14. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{1 \cdot \log \alpha + \color{blue}{1 \cdot \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   15. distribute-rgt-out 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha \cdot \left(1 + 1\right)}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   16. metadata-eval 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\log \alpha \cdot \color{blue}{2}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   17. expm1-udef 98.0%

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

   18. add-exp-log 97.9%

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

   19. sub-neg 97.9%

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

   20. metadata-eval 97.9%

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

5. Applied egg-rr 97.9%

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

6. Taylor expanded in cosTheta around 0 94.2%

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

7. Final simplification 94.2%

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

Alternative 7: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha + single(1.0)) / (single(2.0) * log(alpha))) / (single(pi) / (alpha + single(-1.0)));
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. difference-of-sqr-1 98.2%

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

   2. add-exp-log 98.2%

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

   3. expm1-udef 98.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \alpha\right)}}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. *-commutative 98.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   5. +-commutative 98.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta + 1\right)} \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   6. fma-neg 98.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\left(\left(\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)} \cdot cosTheta\right) \cdot cosTheta + 1\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   7. metadata-eval 98.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\left(\left(\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right) \cdot cosTheta\right) \cdot cosTheta + 1\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   8. associate-*l* 98.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\left(\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)} + 1\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   9. fma-udef 98.2%

      \[\leadsto \frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]

   10. times-frac 98.2%

       \[\leadsto \color{blue}{\frac{\alpha + 1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}} \]

   11. pow2 98.2%

       \[\leadsto \frac{\alpha + 1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), \color{blue}{{cosTheta}^{2}}, 1\right)} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \log \left(\alpha \cdot \alpha\right)} \]

3. Applied egg-rr 98.2%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in cosTheta around 0 94.2%

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

7. Final simplification 94.2%

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

Alternative 8: 65.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * ((single(1.0) / 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. associate-/r* 98.5%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. log-prod 98.6%

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

   5. distribute-rgt-in 98.6%

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

   6. distribute-lft-out 98.6%

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

   7. *-rgt-identity 98.6%

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

   8. *-rgt-identity 98.6%

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

   9. distribute-lft-out 98.6%

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

   10. metadata-eval 98.6%

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

   11. +-commutative 98.6%

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

   12. associate-*l* 98.6%

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

   13. fma-def 98.6%

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

   14. fma-neg 98.6%

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

   15. metadata-eval 98.6%

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

3. Simplified 98.6%

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

   1. metadata-eval 98.6%

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

   2. fma-neg 98.5%

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

   3. difference-of-sqr-1 98.2%

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

   4. add-exp-log 98.2%

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

   5. expm1-udef 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \alpha\right)}}{\log \alpha \cdot \left(\pi \cdot 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\left(\pi \cdot 2\right) \cdot \log \alpha}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   7. associate-*l* 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\pi \cdot \left(2 \cdot \log \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   8. log-pow 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \color{blue}{\log \left({\alpha}^{2}\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   9. pow2 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \log \color{blue}{\left(\alpha \cdot \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   10. *-commutative 98.2%

       \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   11. times-frac 98.0%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + 1}{\log \left(\alpha \cdot \alpha\right)} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   12. log-prod 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha + \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   13. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{1 \cdot \log \alpha} + \log \alpha} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   14. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{1 \cdot \log \alpha + \color{blue}{1 \cdot \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   15. distribute-rgt-out 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha \cdot \left(1 + 1\right)}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   16. metadata-eval 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\log \alpha \cdot \color{blue}{2}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   17. expm1-udef 98.0%

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

   18. add-exp-log 97.9%

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

   19. sub-neg 97.9%

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

   20. metadata-eval 97.9%

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

5. Applied egg-rr 97.9%

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

6. Taylor expanded in cosTheta around 0 94.2%

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

7. Taylor expanded in alpha around 0 65.6%

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

8. Taylor expanded in alpha around inf 65.6%

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

   1. associate-/r* 65.6%

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

   2. log-rec 65.6%

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

10. Simplified 65.6%

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

11. Final simplification 65.6%

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

Alternative 9: 65.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * (single(-1.0) / (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. associate-/r* 98.5%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. log-prod 98.6%

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

   5. distribute-rgt-in 98.6%

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

   6. distribute-lft-out 98.6%

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

   7. *-rgt-identity 98.6%

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

   8. *-rgt-identity 98.6%

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

   9. distribute-lft-out 98.6%

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

   10. metadata-eval 98.6%

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

   11. +-commutative 98.6%

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

   12. associate-*l* 98.6%

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

   13. fma-def 98.6%

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

   14. fma-neg 98.6%

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

   15. metadata-eval 98.6%

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

3. Simplified 98.6%

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

   1. metadata-eval 98.6%

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

   2. fma-neg 98.5%

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

   3. difference-of-sqr-1 98.2%

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

   4. add-exp-log 98.2%

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

   5. expm1-udef 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \alpha\right)}}{\log \alpha \cdot \left(\pi \cdot 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\left(\pi \cdot 2\right) \cdot \log \alpha}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   7. associate-*l* 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\pi \cdot \left(2 \cdot \log \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   8. log-pow 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \color{blue}{\log \left({\alpha}^{2}\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   9. pow2 98.2%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\pi \cdot \log \color{blue}{\left(\alpha \cdot \alpha\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   10. *-commutative 98.2%

       \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   11. times-frac 98.0%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + 1}{\log \left(\alpha \cdot \alpha\right)} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   12. log-prod 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha + \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   13. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{1 \cdot \log \alpha} + \log \alpha} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   14. *-un-lft-identity 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{1 \cdot \log \alpha + \color{blue}{1 \cdot \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   15. distribute-rgt-out 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\log \alpha \cdot \left(1 + 1\right)}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   16. metadata-eval 98.0%

       \[\leadsto \frac{\frac{\alpha + 1}{\log \alpha \cdot \color{blue}{2}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   17. expm1-udef 98.0%

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

   18. add-exp-log 97.9%

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

   19. sub-neg 97.9%

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

   20. metadata-eval 97.9%

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

5. Applied egg-rr 97.9%

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

6. Taylor expanded in cosTheta around 0 94.2%

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

7. Taylor expanded in alpha around 0 65.6%

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

8. Final simplification 65.6%

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

Alternative 10: 65.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * ((single(-1.0) / log(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 94.5%

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

   1. unpow2 94.5%

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

   2. fma-neg 94.4%

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

   3. metadata-eval 94.4%

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

   4. *-lft-identity 94.4%

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

   5. log-pow 94.5%

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

   6. *-commutative 94.5%

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

   7. times-frac 94.3%

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

   8. associate-/r* 94.3%

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

   9. times-frac 94.4%

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

   10. *-commutative 94.4%

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

   11. times-frac 94.4%

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

   12. metadata-eval 94.4%

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

4. Simplified 94.4%

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

5. Taylor expanded in alpha around 0 65.6%

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

6. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2023320 
(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)))))