GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 16.0s

Alternatives: 16

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: 98.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (log (pow (pow alpha 2.0) 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(powf(alpha, 2.0f), ((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(((alpha ^ Float32(2.0)) ^ 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 ^ single(2.0)) ^ single(pi))) * (single(1.0) + (cosTheta * (t_0 * cosTheta))));
end

Derivation

1. Initial program 98.4%

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

   1. add-log-exp 98.3%

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

   2. *-commutative 98.3%

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

   3. exp-to-pow 98.5%

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

   4. pow2 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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))) * log((alpha ^ Float32(Float32(2.0) * Float32(pi))))))
end

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. add-log-exp 98.3%

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

   2. *-commutative 98.3%

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

   3. exp-to-pow 98.5%

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

   4. pow2 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in alpha around 0 98.2%

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

   1. associate-*r* 98.2%

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

   2. *-commutative 98.2%

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

   3. exp-to-pow 98.4%

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

6. Simplified 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

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 98.4%

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

2. Taylor expanded in alpha around 0 98.3%

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

3. Final simplification 98.3%

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

Alternative 4: 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 98.4%

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

2. Final simplification 98.4%

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

Alternative 5: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. fma-neg 98.3%

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

   7. metadata-eval 98.3%

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

   8. *-commutative 98.3%

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

   9. distribute-rgt-neg-out 98.3%

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

   10. distribute-rgt-neg-out 98.3%

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

   11. distribute-lft-neg-in 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.4%

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

   2. div-inv 98.2%

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

   3. pow2 98.2%

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

   4. log-pow 98.4%

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

   5. *-commutative 98.4%

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

5. Applied egg-rr 98.4%

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

6. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

8. Simplified 97.3%

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

   1. metadata-eval 97.3%

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

   2. fma-neg 97.2%

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

   3. difference-of-sqr-1 97.2%

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

   4. *-un-lft-identity 97.2%

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

   5. fma-neg 97.2%

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

   6. metadata-eval 97.2%

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

   7. fma-def 97.2%

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

   8. *-un-lft-identity 97.2%

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

   9. *-un-lft-identity 97.2%

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

   10. times-frac 97.2%

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

10. Applied egg-rr 97.2%

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

    1. /-rgt-identity 97.2%

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

12. Simplified 97.2%

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

    1. un-div-inv 97.3%

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

    2. associate-*r/ 97.5%

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

14. Applied egg-rr 97.5%

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

15. Final simplification 97.5%

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

Alternative 6: 97.6% 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 98.4%

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

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 7: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

   1. metadata-eval 95.1%

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

   2. fma-neg 95.0%

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

   3. difference-of-sqr-1 95.0%

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

   4. *-un-lft-identity 95.0%

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

   5. fma-neg 95.0%

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

   6. metadata-eval 95.0%

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

   7. fma-def 95.0%

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

   8. *-un-lft-identity 95.0%

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

   9. *-commutative 95.0%

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

   10. *-commutative 95.0%

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

   11. frac-times 95.0%

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

   12. associate-*l/ 95.0%

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

   13. *-commutative 95.0%

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

   14. times-frac 94.9%

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

6. Applied egg-rr 94.9%

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

7. Final simplification 94.9%

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

Alternative 8: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

   1. metadata-eval 95.1%

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

   2. fma-neg 95.0%

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

   3. difference-of-sqr-1 95.0%

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

   4. *-un-lft-identity 95.0%

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

   5. fma-neg 95.0%

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

   6. metadata-eval 95.0%

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

   7. fma-def 95.0%

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

   8. *-un-lft-identity 95.0%

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

   9. *-commutative 95.0%

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

   10. *-commutative 95.0%

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

   11. frac-times 95.0%

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

   12. associate-*l/ 95.0%

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

   13. times-frac 95.0%

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

6. Applied egg-rr 95.0%

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

7. Final simplification 95.0%

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

Alternative 9: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

   1. metadata-eval 95.1%

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

   2. fma-neg 95.0%

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

   3. difference-of-sqr-1 95.0%

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

   4. *-un-lft-identity 95.0%

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

   5. fma-neg 95.0%

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

   6. metadata-eval 95.0%

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

   7. fma-def 95.0%

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

   8. *-un-lft-identity 95.0%

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

   9. associate-*r* 95.0%

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

   10. times-frac 94.9%

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

6. Applied egg-rr 94.9%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\pi \cdot 2} \cdot \frac{\alpha + -1}{\log \alpha}} \]
7. Step-by-step derivation

   1. frac-times 95.0%

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

8. Applied egg-rr 95.0%

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

9. Final simplification 95.0%

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

Alternative 10: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

   1. metadata-eval 95.1%

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

   2. fma-neg 95.0%

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

   3. difference-of-sqr-1 95.0%

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

   4. *-un-lft-identity 95.0%

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

   5. fma-neg 95.0%

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

   6. metadata-eval 95.0%

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

   7. fma-def 95.0%

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

   8. *-un-lft-identity 95.0%

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

   9. *-commutative 95.0%

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

   10. *-commutative 95.0%

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

   11. frac-times 95.0%

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

   12. associate-*l/ 95.0%

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

   13. times-frac 95.0%

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

6. Applied egg-rr 95.0%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\log \alpha} \cdot \frac{\frac{\alpha + -1}{\pi}}{2}} \]
7. Step-by-step derivation

   1. associate-*l/ 95.0%

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

   2. associate-/l/ 95.0%

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

8. Simplified 95.0%

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

9. Final simplification 95.0%

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

Alternative 11: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

   1. metadata-eval 95.1%

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

   2. fma-neg 95.0%

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

   3. difference-of-sqr-1 95.0%

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

   4. *-un-lft-identity 95.0%

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

   5. fma-neg 95.0%

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

   6. metadata-eval 95.0%

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

   7. fma-def 95.0%

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

   8. *-un-lft-identity 95.0%

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

   9. associate-*r* 95.0%

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

   10. times-frac 94.9%

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

6. Applied egg-rr 94.9%

   \[\leadsto \color{blue}{\frac{\alpha + 1}{\pi \cdot 2} \cdot \frac{\alpha + -1}{\log \alpha}} \]
7. Step-by-step derivation

   1. clear-num 94.9%

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

   2. frac-times 95.1%

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

   3. *-un-lft-identity 95.1%

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

8. Applied egg-rr 95.1%

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

9. Final simplification 95.1%

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

Alternative 12: 66.7% accurate, 1.1× speedup

Math

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

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(Float32(-0.5) / Float32(Float32(pi) * 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 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. fma-neg 98.3%

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

   7. metadata-eval 98.3%

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

   8. *-commutative 98.3%

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

   9. distribute-rgt-neg-out 98.3%

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

   10. distribute-rgt-neg-out 98.3%

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

   11. distribute-lft-neg-in 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.4%

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

   2. div-inv 98.2%

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

   3. pow2 98.2%

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

   4. log-pow 98.4%

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

   5. *-commutative 98.4%

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

5. Applied egg-rr 98.4%

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

6. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

8. Simplified 97.3%

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

9. Taylor expanded in alpha around 0 65.2%

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

10. Final simplification 65.2%

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

Alternative 13: 66.8% accurate, 1.1× speedup

Math

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

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(Float32(Float32(-0.5) / Float32(pi)) / 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 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. fma-neg 98.3%

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

   7. metadata-eval 98.3%

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

   8. *-commutative 98.3%

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

   9. distribute-rgt-neg-out 98.3%

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

   10. distribute-rgt-neg-out 98.3%

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

   11. distribute-lft-neg-in 98.3%

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

3. Simplified 98.3%

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

   1. associate-/r* 98.4%

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

   2. div-inv 98.2%

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

   3. pow2 98.2%

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

   4. log-pow 98.4%

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

   5. *-commutative 98.4%

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

5. Applied egg-rr 98.4%

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

6. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

8. Simplified 97.3%

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

9. Taylor expanded in alpha around 0 65.2%

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

    1. associate-/r* 65.3%

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

11. Simplified 65.3%

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

12. Final simplification 65.3%

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

Alternative 14: 65.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

5. Taylor expanded in alpha around 0 64.3%

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

   1. associate-/r* 64.3%

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

7. Simplified 64.3%

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

   1. div-inv 64.4%

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

9. Applied egg-rr 64.4%

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

10. Final simplification 64.4%

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

Alternative 15: 65.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

5. Taylor expanded in alpha around 0 64.3%

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

6. Final simplification 64.3%

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

Alternative 16: 65.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 95.1%

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

5. Taylor expanded in alpha around 0 64.3%

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

   1. associate-/r* 64.3%

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

7. Simplified 64.3%

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

8. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2023315 
(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)))))