GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 20.2s

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, 0.6× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (log (pow alpha (* 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(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(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 98.5%

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

   1. add-log-exp 98.5%

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

   2. *-commutative 98.5%

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

   3. exp-to-pow 98.7%

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

   4. pow2 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in alpha around 0 98.5%

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

   1. associate-*r* 98.5%

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

   2. *-commutative 98.5%

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

   3. exp-to-pow 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in alpha around 0 98.5%

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

4. Final simplification 98.5%

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

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. Add Preprocessing

3. Final simplification 98.5%

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

Alternative 4: 97.4% accurate, 1.1× 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. Add Preprocessing

3. Taylor expanded in alpha around 0 97.7%

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

   1. mul-1-neg 97.7%

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

5. Simplified 97.7%

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

6. Final simplification 97.7%

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

Alternative 5: 94.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * (((alpha + single(-1.0)) / log(alpha)) * ((alpha + single(1.0)) / 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. Add Preprocessing

4. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

   3. associate-*r/ 98.3%

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

   4. times-frac 98.1%

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

   5. associate-*r/ 98.1%

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

   6. associate-*r* 98.1%

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

   7. *-commutative 98.1%

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

   8. associate-*r* 98.1%

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

   9. fma-udef 98.1%

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

   10. *-commutative 98.1%

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

   11. fma-def 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 95.4%

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

   1. times-frac 95.3%

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

   2. +-commutative 95.3%

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

   3. sub-neg 95.3%

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

   4. metadata-eval 95.3%

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

9. Simplified 95.3%

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

10. Final simplification 95.3%

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

Alternative 6: 94.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * (((alpha + single(1.0)) / log(alpha)) * ((alpha + single(-1.0)) / 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. Add Preprocessing

4. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

   3. associate-*r/ 98.3%

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

   4. times-frac 98.1%

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

   5. associate-*r/ 98.1%

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

   6. associate-*r* 98.1%

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

   7. *-commutative 98.1%

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

   8. associate-*r* 98.1%

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

   9. fma-udef 98.1%

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

   10. *-commutative 98.1%

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

   11. fma-def 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 95.4%

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

   1. times-frac 95.3%

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

   2. +-commutative 95.3%

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

   3. sub-neg 95.3%

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

   4. metadata-eval 95.3%

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

9. Simplified 95.3%

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

    1. *-commutative 95.3%

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

    2. clear-num 95.3%

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

    3. frac-times 95.3%

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

    4. *-un-lft-identity 95.3%

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

11. Applied egg-rr 95.3%

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

    1. *-commutative 95.3%

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

    2. associate-/r* 95.4%

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

    3. *-lft-identity 95.4%

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

    4. metadata-eval 95.4%

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

    5. times-frac 95.4%

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

    6. neg-mul-1 95.4%

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

    7. neg-mul-1 95.4%

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

    8. associate-/r* 95.3%

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

    9. associate-*r/ 95.1%

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

    10. distribute-lft-neg-out 95.1%

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

    11. distribute-rgt-neg-out 95.1%

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

    12. associate-/l* 95.4%

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

    13. distribute-rgt-neg-out 95.4%

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

    14. distribute-lft-neg-out 95.4%

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

    15. times-frac 95.3%

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

13. Simplified 95.4%

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

14. Final simplification 95.4%

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

Alternative 7: 94.9% accurate, 1.1× 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. Add Preprocessing

4. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

   3. associate-*r/ 98.3%

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

   4. times-frac 98.1%

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

   5. associate-*r/ 98.1%

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

   6. associate-*r* 98.1%

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

   7. *-commutative 98.1%

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

   8. associate-*r* 98.1%

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

   9. fma-udef 98.1%

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

   10. *-commutative 98.1%

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

   11. fma-def 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 95.4%

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

8. Final simplification 95.4%

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

Alternative 8: 94.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * ((alpha + single(-1.0)) / (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. Add Preprocessing

4. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

   3. associate-*r/ 98.3%

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

   4. times-frac 98.1%

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

   5. associate-*r/ 98.1%

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

   6. associate-*r* 98.1%

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

   7. *-commutative 98.1%

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

   8. associate-*r* 98.1%

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

   9. fma-udef 98.1%

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

   10. *-commutative 98.1%

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

   11. fma-def 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 95.4%

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

   1. times-frac 95.3%

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

   2. +-commutative 95.3%

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

   3. sub-neg 95.3%

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

   4. metadata-eval 95.3%

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

9. Simplified 95.3%

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

    1. clear-num 95.3%

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

    2. frac-times 95.5%

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

    3. *-un-lft-identity 95.5%

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

11. Applied egg-rr 95.5%

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

12. Final simplification 95.5%

    \[\leadsto 0.5 \cdot \frac{\alpha + -1}{\log \alpha \cdot \frac{\pi}{\alpha + 1}} \]
13. Add Preprocessing

Alternative 9: 94.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * (((alpha + single(-1.0)) / 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. Add Preprocessing

4. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

   3. associate-*r/ 98.3%

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

   4. times-frac 98.1%

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

   5. associate-*r/ 98.1%

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

   6. associate-*r* 98.1%

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

   7. *-commutative 98.1%

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

   8. associate-*r* 98.1%

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

   9. fma-udef 98.1%

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

   10. *-commutative 98.1%

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

   11. fma-def 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 95.4%

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

   1. times-frac 95.3%

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

   2. +-commutative 95.3%

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

   3. sub-neg 95.3%

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

   4. metadata-eval 95.3%

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

9. Simplified 95.3%

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

    1. add-cube-cbrt 94.6%

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

    2. log-prod 94.6%

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

    3. pow2 94.6%

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

11. Applied egg-rr 94.6%

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

    1. log-pow 94.7%

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

    2. distribute-lft1-in 94.7%

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

    3. metadata-eval 94.7%

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

13. Simplified 94.7%

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

    1. associate-*r/ 94.7%

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

    2. add-log-exp 94.7%

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

    3. *-commutative 94.7%

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

    4. exp-to-pow 94.7%

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

    5. pow3 94.6%

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

    6. add-cube-cbrt 95.2%

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

    7. clear-num 95.2%

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

    8. *-un-lft-identity 95.2%

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

    9. frac-times 95.2%

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

    10. clear-num 95.2%

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

    11. *-commutative 95.2%

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

    12. associate-/r* 95.4%

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

15. Applied egg-rr 95.5%

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

16. Final simplification 95.5%

    \[\leadsto 0.5 \cdot \frac{\frac{\alpha + -1}{\log \alpha}}{\frac{\pi}{\alpha + 1}} \]
17. Add Preprocessing

Alternative 10: 65.5% 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. Add Preprocessing

4. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

   3. associate-*r/ 98.3%

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

   4. times-frac 98.1%

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

   5. associate-*r/ 98.1%

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

   6. associate-*r* 98.1%

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

   7. *-commutative 98.1%

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

   8. associate-*r* 98.1%

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

   9. fma-udef 98.1%

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

   10. *-commutative 98.1%

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

   11. fma-def 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 95.4%

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

   1. times-frac 95.3%

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

   2. +-commutative 95.3%

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

   3. sub-neg 95.3%

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

   4. metadata-eval 95.3%

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

9. Simplified 95.3%

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

10. Taylor expanded in alpha around 0 64.7%

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

11. Final simplification 64.7%

    \[\leadsto 0.5 \cdot \frac{-1}{\pi \cdot \log \alpha} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))))