GTR1 distribution

Percentage Accurate: 98.5% → 98.4%

Time: 11.4s

Alternatives: 8

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: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\alpha}^{2} + -1\\ \frac{\frac{1}{\log \left({\alpha}^{2}\right)}}{\frac{\pi}{t\_0} \cdot \mathsf{fma}\left(t\_0, {cosTheta}^{2}, 1\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	t_0 = Float32((alpha ^ Float32(2.0)) + Float32(-1.0))
	return Float32(Float32(Float32(1.0) / log((alpha ^ Float32(2.0)))) / Float32(Float32(Float32(pi) / t_0) * fma(t_0, (cosTheta ^ Float32(2.0)), Float32(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
3. Step-by-step derivation

   1. associate-/l/ 98.6%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. +-commutative 98.5%

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

   5. associate-*l* 98.5%

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

   6. fma-undefine 98.5%

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

   7. fma-neg 98.5%

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

   8. metadata-eval 98.5%

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

   9. *-commutative 98.5%

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

   10. associate-/l/ 98.3%

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

   11. div-inv 98.2%

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

   12. clear-num 98.2%

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

   13. associate-*l/ 98.5%

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

4. Applied egg-rr 98.6%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t\_0}{\log \left({\alpha}^{\left(2 \cdot \pi\right)}\right) \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot t\_0\right)\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = single(-1.0) + (alpha * alpha);
	tmp = t_0 / (log((alpha ^ (single(2.0) * single(pi)))) * (single(1.0) + (cosTheta * (cosTheta * t_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
3. Step-by-step derivation

   1. *-commutative 98.5%

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

   2. add-log-exp 98.5%

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

   3. exp-to-pow 98.6%

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

   4. pow2 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in alpha around 0 98.4%

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

   1. associate-*r* 98.4%

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

   2. *-commutative 98.4%

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

   3. exp-to-pow 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t\_0}{\left(1 + cosTheta \cdot \left(cosTheta \cdot t\_0\right)\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = single(-1.0) + (alpha * alpha);
	tmp = t_0 / ((single(1.0) + (cosTheta * (cosTheta * t_0))) * (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%

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

Alternative 4: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) + (alpha * alpha)) / ((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 98.2%

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

4. Final simplification 98.2%

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

Alternative 5: 94.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (((single(1.0) + alpha) * (alpha + single(-1.0))) / 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. Step-by-step derivation

   1. associate-/l/ 98.6%

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

   2. associate-/r* 98.4%

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

   3. fma-neg 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

   6. associate-*l* 98.3%

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

   7. fma-define 98.3%

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

   8. fma-neg 98.3%

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

   9. metadata-eval 98.3%

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

3. Simplified 98.3%

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

   1. metadata-eval 98.3%

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

   2. fma-neg 98.4%

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

   3. associate-/l/ 98.4%

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

   4. difference-of-sqr-1 98.2%

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

   5. metadata-eval 98.2%

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

   6. fma-neg 98.2%

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

   7. fma-undefine 98.1%

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

   8. associate-*l* 98.2%

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

   9. +-commutative 98.2%

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

   10. times-frac 98.3%

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

   11. sub-neg 98.3%

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

   12. metadata-eval 98.3%

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

   13. +-commutative 98.3%

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

   14. associate-*l* 98.3%

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

   15. fma-undefine 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in cosTheta around 0 95.7%

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

8. Final simplification 95.7%

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

Alternative 6: 65.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. associate-/l/ 98.6%

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

   2. associate-/r* 98.4%

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

   3. fma-neg 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

   6. associate-*l* 98.3%

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

   7. fma-define 98.3%

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

   8. fma-neg 98.3%

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

   9. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in alpha around 0 68.8%

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

6. Taylor expanded in cosTheta around 0 67.4%

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

   1. div-inv 67.4%

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

   2. pow2 67.4%

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

   3. log-pow 67.4%

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

8. Applied egg-rr 67.4%

   \[\leadsto \color{blue}{\frac{-1}{\pi} \cdot \frac{1}{2 \cdot \log \alpha}} \]
9. Add Preprocessing

Alternative 7: 65.2% accurate, 1.2× 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 98.5%

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

   1. associate-/l/ 98.6%

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

   2. associate-/r* 98.4%

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

   3. fma-neg 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

   6. associate-*l* 98.3%

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

   7. fma-define 98.3%

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

   8. fma-neg 98.3%

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

   9. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in alpha around 0 68.8%

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

6. Taylor expanded in cosTheta around 0 67.4%

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

7. Taylor expanded in alpha around 0 67.3%

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

   1. associate-/r* 67.4%

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

9. Simplified 67.4%

   \[\leadsto \color{blue}{\frac{\frac{-0.5}{\pi}}{\log \alpha}} \]
10. Add Preprocessing

Alternative 8: 65.2% accurate, 1.2× 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 98.5%

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

   1. associate-/l/ 98.6%

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

   2. associate-/r* 98.4%

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

   3. fma-neg 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

   6. associate-*l* 98.3%

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

   7. fma-define 98.3%

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

   8. fma-neg 98.3%

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

   9. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in alpha around 0 68.8%

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

6. Taylor expanded in cosTheta around 0 67.4%

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

7. Taylor expanded in alpha around 0 67.3%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \log \alpha}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))