GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 10.2s

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: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (log (pow (* alpha alpha) 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 * alpha), ((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((Float32(alpha * alpha) ^ 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 * alpha) ^ 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. Step-by-step derivation

   1. add-log-exp 98.4%

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

   2. *-commutative 98.4%

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

   3. exp-to-pow 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Final simplification 98.5%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0 97.2%

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

4. Simplified 97.2%

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

5. Final simplification 97.2%

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

Alternative 4: 95.1% accurate, 1.0× 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. Step-by-step derivation

   1. associate-/r* 98.5%

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

   2. difference-of-sqr-1 98.1%

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

   3. *-commutative 98.1%

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

   4. times-frac 98.1%

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

   5. *-commutative 98.1%

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

   6. times-frac 98.1%

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

   7. difference-of-sqr-1 98.5%

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

   8. associate-/l/ 98.4%

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

   9. log-prod 98.3%

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

   10. count-2 98.3%

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

   11. *-commutative 98.3%

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

   12. fma-neg 98.4%

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

   13. metadata-eval 98.4%

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

   14. +-commutative 98.4%

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

3. Simplified 98.4%

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

   1. fma-udef 98.3%

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

   2. difference-of-sqr--1 98.2%

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

   3. *-un-lft-identity 98.2%

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

   4. times-frac 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in cosTheta around 0 95.0%

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

   1. times-frac 95.1%

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

   2. +-commutative 95.1%

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

   3. sub-neg 95.1%

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

   4. metadata-eval 95.1%

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

10. Simplified 95.1%

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

11. Final simplification 95.1%

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

Alternative 5: 47.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. difference-of-sqr-1 98.1%

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

   2. associate-*l* 98.1%

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

   3. times-frac 98.0%

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

   4. pow2 98.0%

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

   5. log-pow 98.1%

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

   6. *-commutative 98.1%

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

   7. +-commutative 98.1%

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

   8. fma-neg 98.1%

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

   9. metadata-eval 98.1%

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

   10. associate-*l* 98.1%

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

   11. fma-udef 98.1%

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

3. Applied egg-rr 98.1%

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

4. Taylor expanded in cosTheta around 0 94.9%

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

   1. *-commutative 94.9%

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

6. Simplified 94.9%

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

7. Taylor expanded in alpha around 0 46.1%

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

8. Final simplification 46.1%

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

Alternative 6: 66.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-0.5) / (log(alpha) * (single(pi) * (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. Step-by-step derivation

   1. associate-/r* 98.5%

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

   2. difference-of-sqr-1 98.1%

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

   3. *-commutative 98.1%

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

   4. times-frac 98.1%

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

   5. *-commutative 98.1%

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

   6. times-frac 98.1%

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

   7. difference-of-sqr-1 98.5%

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

   8. associate-/l/ 98.4%

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

   9. log-prod 98.3%

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

   10. count-2 98.3%

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

   11. *-commutative 98.3%

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

   12. fma-neg 98.4%

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

   13. metadata-eval 98.4%

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

   14. +-commutative 98.4%

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

3. Simplified 98.4%

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

   1. fma-udef 98.3%

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

   2. difference-of-sqr--1 98.2%

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

   3. *-un-lft-identity 98.2%

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

   4. times-frac 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in alpha around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. mul-1-neg 64.6%

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

   3. sub-neg 64.6%

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

   4. unpow2 64.6%

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

10. Simplified 64.6%

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

11. Final simplification 64.6%

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

Alternative 7: 46.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha + single(1.0)) / single(pi)) * (single(-0.5) / 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. difference-of-sqr-1 98.1%

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

   2. associate-*l* 98.1%

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

   3. times-frac 98.0%

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

   4. pow2 98.0%

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

   5. log-pow 98.1%

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

   6. *-commutative 98.1%

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

   7. +-commutative 98.1%

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

   8. fma-neg 98.1%

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

   9. metadata-eval 98.1%

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

   10. associate-*l* 98.1%

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

   11. fma-udef 98.1%

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

3. Applied egg-rr 98.1%

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

4. Taylor expanded in cosTheta around 0 94.9%

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

   1. *-commutative 94.9%

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

6. Simplified 94.9%

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

7. Taylor expanded in alpha around 0 45.5%

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

8. Final simplification 45.5%

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

Alternative 8: -0.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right) \cdot \left(\pi \cdot \frac{0}{0}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/ t_0 (* (+ 1.0 (* cosTheta (* t_0 cosTheta))) (* PI (/ 0.0 0.0))))))

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) * (0.0f / 0.0f)));
}

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) * Float32(Float32(0.0) / Float32(0.0)))))
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) * (single(0.0) / single(0.0))));
end

Derivation

1. Initial program 98.5%

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

   1. log-prod 98.4%

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

   2. flip-+ -0.0%

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

   3. pow2 -0.0%

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

   4. pow2 -0.0%

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

3. Applied egg-rr -0.0%

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

5. Simplified -0.0%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \color{blue}{\frac{0}{0}}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

6. Final simplification -0.0%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right)\right) \cdot \left(\pi \cdot \frac{0}{0}\right)} \]

Reproduce

herbie shell --seed 2023230 
(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)))))