GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 13.0s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. add-log-exp 98.3%

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

   2. *-commutative 98.3%

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

   3. exp-to-pow 98.7%

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

   4. pow2 98.7%

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

5. Applied egg-rr 98.7%

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

6. Taylor expanded in alpha around 0 98.4%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ -1.0 (* alpha alpha))))
   (/
    t_0
    (* (log (pow (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(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(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 98.3%

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

   1. add-log-exp 98.3%

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

   2. *-commutative 98.3%

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

   3. exp-to-pow 98.7%

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

   4. pow2 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

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

   2. difference-of-sqr--1 98.0%

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

   3. add-exp-log 98.1%

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

   4. expm1-udef 98.1%

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

   5. *-commutative 98.1%

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

   6. times-frac 98.2%

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

   7. pow2 98.2%

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

   8. log-pow 98.3%

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

   9. *-commutative 98.3%

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

   10. expm1-udef 98.3%

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

   11. add-exp-log 98.3%

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

   12. sub-neg 98.3%

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

   13. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. frac-times 98.1%

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

   2. *-commutative 98.1%

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

   3. metadata-eval 98.1%

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

   4. sub-neg 98.1%

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

   5. difference-of-sqr--1 98.3%

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

   6. fma-udef 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-*l* 98.4%

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

   9. associate-/r* 98.4%

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

   10. *-commutative 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	float t_0 = -1.0f + (alpha * alpha);
	return t_0 / ((1.0f + (cosTheta * (cosTheta * t_0))) * (2.0f * (((float) M_PI) * logf(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(2.0) * Float32(Float32(pi) * log(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(2.0) * (single(pi) * log(alpha))));
end

Derivation

1. Initial program 98.3%

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

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%

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

Alternative 5: 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 98.3%

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

2. Final simplification 98.3%

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

Alternative 6: 97.7% accurate, 1.0× 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 98.3%

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

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%

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

Alternative 7: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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. Step-by-step derivation

   1. difference-of-sqr-1 97.1%

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

   2. sub-neg 97.1%

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

   3. metadata-eval 97.1%

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

   4. pow2 97.1%

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

   5. pow-to-exp 97.1%

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

   6. add-log-exp 97.2%

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

   7. associate-*l* 97.1%

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

   8. times-frac 97.2%

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

   9. *-commutative 97.2%

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

   10. +-commutative 97.2%

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

   11. fma-def 97.2%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 94.0%

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

   14. sqr-neg 94.0%

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

   15. sqrt-unprod 94.0%

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

   16. add-sqr-sqrt 94.0%

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

6. Applied egg-rr 94.0%

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

7. Taylor expanded in cosTheta around 0 94.3%

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

8. Final simplification 94.3%

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

Alternative 8: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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. Step-by-step derivation

   1. difference-of-sqr-1 97.1%

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

   2. sub-neg 97.1%

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

   3. metadata-eval 97.1%

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

   4. pow2 97.1%

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

   5. pow-to-exp 97.1%

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

   6. add-log-exp 97.2%

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

   7. associate-*l* 97.1%

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

   8. times-frac 97.2%

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

   9. *-commutative 97.2%

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

   10. +-commutative 97.2%

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

   11. fma-def 97.2%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 94.0%

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

   14. sqr-neg 94.0%

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

   15. sqrt-unprod 94.0%

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

   16. add-sqr-sqrt 94.0%

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

6. Applied egg-rr 94.0%

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

7. Taylor expanded in cosTheta around 0 94.3%

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

8. Final simplification 94.3%

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

Alternative 9: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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. Step-by-step derivation

   1. difference-of-sqr-1 97.1%

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

   2. sub-neg 97.1%

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

   3. metadata-eval 97.1%

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

   4. pow2 97.1%

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

   5. pow-to-exp 97.1%

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

   6. add-log-exp 97.2%

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

   7. associate-*l* 97.1%

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

   8. times-frac 97.2%

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

   9. *-commutative 97.2%

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

   10. +-commutative 97.2%

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

   11. fma-def 97.2%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 94.0%

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

   14. sqr-neg 94.0%

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

   15. sqrt-unprod 94.0%

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

   16. add-sqr-sqrt 94.0%

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

6. Applied egg-rr 94.0%

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

   1. clear-num 94.0%

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

   2. frac-times 94.2%

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

   3. *-un-lft-identity 94.2%

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

   4. associate-*l* 94.2%

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

8. Applied egg-rr 94.2%

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

9. Taylor expanded in cosTheta around 0 94.5%

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

10. Final simplification 94.5%

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

Alternative 10: 65.9% 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 98.3%

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

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. Step-by-step derivation

   1. difference-of-sqr-1 97.1%

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

   2. sub-neg 97.1%

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

   3. metadata-eval 97.1%

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

   4. pow2 97.1%

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

   5. pow-to-exp 97.1%

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

   6. add-log-exp 97.2%

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

   7. associate-*l* 97.1%

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

   8. times-frac 97.2%

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

   9. *-commutative 97.2%

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

   10. +-commutative 97.2%

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

   11. fma-def 97.2%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 94.0%

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

   14. sqr-neg 94.0%

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

   15. sqrt-unprod 94.0%

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

   16. add-sqr-sqrt 94.0%

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

6. Applied egg-rr 94.0%

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

7. Taylor expanded in cosTheta around 0 94.3%

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

8. Taylor expanded in alpha around 0 67.0%

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

9. Final simplification 67.0%

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

Reproduce

herbie shell --seed 2023319 
(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)))))