GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 8.4s

Alternatives: 9

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. *-commutative N/A

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

   12. log-pow-rev N/A

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

   13. pow2 N/A

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

   14. log-pow-rev N/A

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

3. Applied rewrites 98.6%

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

Alternative 2: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. associate-*l* N/A

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

   7. difference-of-sqr-1 N/A

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

   8. difference-of-sqr--1-rev N/A

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

   9. pow2 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-out N/A

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

   14. +-commutative N/A

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

   15. associate-+l+ N/A

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

   16. unpow2 N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

   19. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(\left(-1 \cdot cosTheta\right) \cdot cosTheta + \left({cosTheta}^{2} \cdot {\alpha}^{2} + \color{blue}{{\left({\alpha}^{2}\right)}^{0}}\right)\right)} \]

   20. metadata-eval N/A

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

   21. pow-plus N/A

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

   22. inv-pow N/A

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

3. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. difference-of-sqr-1 N/A

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

   4. difference-of-sqr--1-rev N/A

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

   5. lower-fma.f32 98.4

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-log.f32 N/A

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

   10. pow2 N/A

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

   11. log-pow-rev N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-log.f32 98.5

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

   20. lift-+.f32 N/A

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

   21. lift-*.f32 N/A

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

3. Applied rewrites 98.5%

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

Alternative 4: 97.6% accurate, 1.3× 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

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

   1. metadata-eval N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 97.6

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

4. Applied rewrites 97.6%

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

Alternative 5: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. metadata-eval N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 97.6

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

4. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. difference-of-sqr-1 N/A

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

   4. difference-of-sqr--1 N/A

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

   5. lift-fma.f32 97.5

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

6. Applied rewrites 97.6%

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

Alternative 6: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(Float32(pi) + Float32(pi)) * Float32(Float32(Float32(1.0) - Float32(cosTheta * cosTheta)) * 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. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. difference-of-sqr-1 N/A

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

   4. difference-of-sqr--1-rev N/A

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

   5. lower-fma.f32 98.4

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-log.f32 N/A

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

   10. pow2 N/A

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

   11. log-pow-rev N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-log.f32 98.5

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

   20. lift-+.f32 N/A

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

   21. lift-*.f32 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-fma.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. pow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. difference-of-sqr--1 N/A

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

   9. difference-of-sqr-1 N/A

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

   10. *-commutative N/A

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

   11. lower-+.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. difference-of-sqr-1 N/A

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

   14. difference-of-sqr--1 N/A

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

   15. lower-*.f32 N/A

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

   16. lift-fma.f32 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. count-2 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. unpow-prod-down N/A

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

   13. mul-1-neg N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. lift--.f32 N/A

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

   17. lift-*.f32 N/A

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

   18. lift-log.f32 97.6

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

8. Applied rewrites 97.6%

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

Alternative 7: 95.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.2%

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

Alternative 8: 95.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. pow2 N/A

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

   3. difference-of-sqr-1 N/A

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

   4. difference-of-sqr--1-rev N/A

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

   5. lower-fma.f32 N/A

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

   6. log-pow-rev N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-log.f32 95.3

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

4. Applied rewrites 95.3%

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

Alternative 9: 65.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.1%

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

4. Taylor expanded in cosTheta around 0

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

   1. lift--.f32 94.9

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

6. Applied rewrites 94.9%

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

7. Taylor expanded in alpha around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 47.3

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

9. Applied rewrites 47.3%

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

10. Taylor expanded in alpha around 0

    \[\leadsto \frac{\frac{1}{2}}{\log \alpha \cdot \pi} \cdot -1 \]
11. Step-by-step derivation

12. Applied rewrites 65.9%

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

Reproduce

herbie shell --seed 2025134 
(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)))))