GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 14.6s

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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

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 + (cosTheta * (cosTheta * t_0))));
}

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(cosTheta * Float32(cosTheta * t_0)))))
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) + (cosTheta * (cosTheta * t_0))));
end

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{\frac{0.5}{\log \alpha} \cdot t\_0}{\pi \cdot \left(1 + t\_0 \cdot \left(cosTheta \cdot cosTheta\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    (* (/ 0.5 (log alpha)) t_0)
    (* PI (+ 1.0 (* t_0 (* cosTheta cosTheta)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) + -1.0f;
	return ((0.5f / logf(alpha)) * t_0) / (((float) M_PI) * (1.0f + (t_0 * (cosTheta * cosTheta))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) + Float32(-1.0))
	return Float32(Float32(Float32(Float32(0.5) / log(alpha)) * t_0) / Float32(Float32(pi) * Float32(Float32(1.0) + Float32(t_0 * Float32(cosTheta * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) + single(-1.0);
	tmp = ((single(0.5) / log(alpha)) * t_0) / (single(pi) * (single(1.0) + (t_0 * (cosTheta * cosTheta))));
end

Derivation

1. Initial program 98.4%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right)}\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\alpha \cdot \alpha - 1\right), \log \left(\alpha \cdot \alpha\right)\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\alpha \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right), \log \left(\alpha \cdot \alpha\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), \left(\mathsf{neg}\left(1\right)\right)\right), \log \left(\alpha \cdot \alpha\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), \left(\mathsf{neg}\left(1\right)\right)\right), \log \left(\alpha \cdot \alpha\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \log \left(\alpha \cdot \alpha\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   10. pow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \log \left({\alpha}^{2}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   11. pow-to-exp N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \log \left(e^{\log \alpha \cdot 2}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   12. rem-log-exp N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \left(\log \alpha \cdot 2\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(\log \alpha, 2\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   14. log-lowering-log.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)\right) \]

   16. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \left(\color{blue}{1} + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)\right) \]

   17. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), 2\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)\right)\right) \]

4. Applied egg-rr 98.3%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{\frac{\log \alpha \cdot 2}{\alpha \cdot \alpha + -1}}\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{PI.f32}\left(\right)}, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{\log \alpha \cdot 2} \cdot \left(\alpha \cdot \alpha + -1\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{PI.f32}\left(\right)}, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{\log \alpha \cdot 2}\right), \left(\alpha \cdot \alpha + -1\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{PI.f32}\left(\right)}, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{2 \cdot \log \alpha}\right), \left(\alpha \cdot \alpha + -1\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{\frac{1}{2}}{\log \alpha}\right), \left(\alpha \cdot \alpha + -1\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{\frac{1}{2}}{\log \alpha}\right), \left(\alpha \cdot \alpha + -1\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \log \alpha\right), \left(\alpha \cdot \alpha + -1\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   8. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{log.f32}\left(\alpha\right)\right), \left(\alpha \cdot \alpha + -1\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), -1\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right)\right)\right) \]

6. Applied egg-rr 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. div-sub N/A

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

   2. --lowering--.f32 N/A

      \[\leadsto \mathsf{\_.f32}\left(\left(\frac{\alpha \cdot \alpha}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right), \color{blue}{\left(\frac{1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)}\right) \]

4. Applied egg-rr 98.4%

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

   1. sub-div N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. associate-/l/ N/A

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

   5. frac-2neg N/A

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

   6. div-inv N/A

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

   7. distribute-neg-frac2 N/A

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

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{\alpha \cdot \alpha + -1}{1 + \left(\alpha \cdot \alpha + -1\right) \cdot \left(cosTheta \cdot cosTheta\right)}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot 2\right)}\right)\right)}\right) \]

6. Applied egg-rr 98.2%

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

Alternative 4: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\log \alpha, \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(\color{blue}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + \left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 - {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left({cosTheta}^{2}\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left(cosTheta \cdot cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{2}\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{2}\right)\right)\right) \]

   13. PI-lowering-PI.f32 97.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), 2\right)\right)\right) \]

5. Simplified 97.8%

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

6. Final simplification 97.8%

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

Alternative 5: 97.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right) \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\log \alpha, \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(\color{blue}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + -1 \cdot {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 + \left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \left(1 - {cosTheta}^{2}\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left({cosTheta}^{2}\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \left(cosTheta \cdot cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{2}\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{2}\right)\right)\right) \]

   13. PI-lowering-PI.f32 97.8%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{log.f32}\left(\alpha\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), 2\right)\right)\right) \]

5. Simplified 97.8%

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. associate-*l* N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\alpha \cdot \alpha + -1}{\log \alpha}\right), \color{blue}{\left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\alpha \cdot \alpha + -1\right), \log \alpha\right), \left(\color{blue}{\left(1 - cosTheta \cdot cosTheta\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\alpha \cdot \alpha\right), -1\right), \log \alpha\right), \left(\left(\color{blue}{1} - cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \log \alpha\right), \left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \]

   9. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \left(\left(1 - \color{blue}{cosTheta \cdot cosTheta}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{*.f32}\left(\left(1 - cosTheta \cdot cosTheta\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)\right) \]

   11. --lowering--.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \left(cosTheta \cdot cosTheta\right)\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right), \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   14. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right), \mathsf{/.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   16. PI-lowering-PI.f32 97.7%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), -1\right), \mathsf{log.f32}\left(\alpha\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(cosTheta, cosTheta\right)\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \frac{1}{2}\right)\right)\right) \]

7. Applied egg-rr 97.7%

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

9. Applied egg-rr 97.6%

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

10. Final simplification 97.6%

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

Alternative 6: 95.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right)}\right), \color{blue}{\log \left({\alpha}^{2}\right)}\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\alpha}^{2} - 1\right), \mathsf{PI}\left(\right)\right), \log \color{blue}{\left({\alpha}^{2}\right)}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\alpha}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right), \mathsf{PI}\left(\right)\right), \log \left({\color{blue}{\alpha}}^{2}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({\alpha}^{2} + -1\right), \mathsf{PI}\left(\right)\right), \log \left({\alpha}^{2}\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 + {\alpha}^{2}\right), \mathsf{PI}\left(\right)\right), \log \left({\color{blue}{\alpha}}^{2}\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \left({\alpha}^{2}\right)\right), \mathsf{PI}\left(\right)\right), \log \left({\color{blue}{\alpha}}^{2}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \left(\alpha \cdot \alpha\right)\right), \mathsf{PI}\left(\right)\right), \log \left({\alpha}^{2}\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{PI}\left(\right)\right), \log \left({\alpha}^{2}\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{PI.f32}\left(\right)\right), \log \left({\alpha}^{\color{blue}{2}}\right)\right) \]

   11. log-lowering-log.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{log.f32}\left(\left({\alpha}^{2}\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{log.f32}\left(\left(\alpha \cdot \alpha\right)\right)\right) \]

   13. *-lowering-*.f32 95.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(\alpha, \alpha\right)\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right) \]

5. Simplified 95.6%

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

6. Final simplification 95.6%

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

Alternative 7: 95.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\log \left({\alpha}^{2}\right)}\right)\right) \]

   2. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \log \color{blue}{\left({\alpha}^{2}\right)}\right)\right) \]

   3. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\left({\alpha}^{2}\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\left(\alpha \cdot \alpha\right)\right)\right)\right) \]

   5. *-lowering-*.f32 95.5%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right), 1\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\mathsf{*.f32}\left(\alpha, \alpha\right)\right)\right)\right) \]

5. Simplified 95.5%

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

6. Final simplification 95.5%

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

Alternative 8: 65.8% 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 98.4%

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

   1. div-sub N/A

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

   2. --lowering--.f32 N/A

      \[\leadsto \mathsf{\_.f32}\left(\left(\frac{\alpha \cdot \alpha}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right), \color{blue}{\left(\frac{1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}\right)}\right) \]

4. Applied egg-rr 98.4%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \color{blue}{\left(-1 \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}\right) \]
8. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \left(-1 \cdot 1 + \color{blue}{-1 \cdot \left(-1 \cdot {cosTheta}^{2}\right)}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \left(-1 + \color{blue}{-1} \cdot \left(-1 \cdot {cosTheta}^{2}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \left(-1 + \left(-1 \cdot -1\right) \cdot \color{blue}{{cosTheta}^{2}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \left(-1 + 1 \cdot {\color{blue}{cosTheta}}^{2}\right)\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \left(-1 + {cosTheta}^{\color{blue}{2}}\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \mathsf{+.f32}\left(-1, \color{blue}{\left({cosTheta}^{2}\right)}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \mathsf{+.f32}\left(-1, \left(cosTheta \cdot \color{blue}{cosTheta}\right)\right)\right) \]

   8. *-lowering-*.f32 68.0%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right), \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(cosTheta, \color{blue}{cosTheta}\right)\right)\right) \]

9. Simplified 68.0%

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

10. Taylor expanded in cosTheta around 0

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

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)}\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\log \alpha}\right)\right) \]

    3. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \log \color{blue}{\alpha}\right)\right) \]

    4. log-lowering-log.f32 66.7%

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{log.f32}\left(\alpha\right)\right)\right) \]

12. Simplified 66.7%

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

Reproduce

herbie shell --seed 2024164 
(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)))))