GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 4.2s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. lower-fma.f32 N/A

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

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

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

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

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

   12. pow2 N/A

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

   13. metadata-eval N/A

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

   14. negate-sub N/A

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

   15. lower-*.f32 N/A

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

   16. negate-sub N/A

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

   17. metadata-eval N/A

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

   18. pow2 N/A

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

   19. lift-fma.f32 98.5

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

5. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	return fmaf(alpha, alpha, -1.0f) / (((logf(alpha) * ((float) M_PI)) * 2.0f) * 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(log(alpha) * Float32(pi)) * Float32(2.0)) * 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. pow2 N/A

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

   4. negate-sub N/A

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

   5. pow2 N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 98.4

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-log.f32 N/A

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

   12. log-pow-rev N/A

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

   13. pow2 N/A

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

   14. pow-pow N/A

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

   15. log-pow-rev N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f32 N/A

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

   21. lower-log.f32 N/A

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

   22. lift-PI.f32 98.5

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

3. Applied rewrites 98.5%

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

Alternative 5: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 97.5

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

4. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lower-fma.f32 97.4

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-log.f32 N/A

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

   10. log-pow-rev N/A

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

   11. unpow-prod-down N/A

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

   12. pow2 N/A

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

   13. log-pow-rev N/A

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

   14. log-pow-rev N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

   19. lift-log.f32 N/A

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

   20. lift-PI.f32 97.5

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. log-pow-rev N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. unpow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

9. Applied rewrites 97.5%

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

    1. lift-*.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-log.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. lift--.f32 N/A

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

    7. *-commutative N/A

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

    8. pow2 N/A

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

    9. negate-sub N/A

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

    10. mul-1-neg N/A

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

    11. associate-*r* N/A

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

    12. *-commutative N/A

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

    13. lower-*.f32 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f32 N/A

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

    16. mul-1-neg N/A

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

    17. negate-sub N/A

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

    18. pow2 N/A

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

    19. lift--.f32 N/A

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

    20. lift-*.f32 N/A

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

    21. lift-log.f32 N/A

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

    22. lift-PI.f32 97.5

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

11. Applied rewrites 97.5%

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

Alternative 6: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 97.5

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

4. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lower-fma.f32 97.4

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-log.f32 N/A

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

   10. log-pow-rev N/A

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

   11. unpow-prod-down N/A

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

   12. pow2 N/A

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

   13. log-pow-rev N/A

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

   14. log-pow-rev N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

   19. lift-log.f32 N/A

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

   20. lift-PI.f32 97.5

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. log-pow-rev N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. unpow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

9. Applied rewrites 97.5%

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

Alternative 7: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 97.5

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

4. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lower-fma.f32 97.4

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-log.f32 N/A

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

   10. log-pow-rev N/A

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

   11. unpow-prod-down N/A

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

   12. pow2 N/A

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

   13. log-pow-rev N/A

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

   14. log-pow-rev N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

   19. lift-log.f32 N/A

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

   20. lift-PI.f32 97.5

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. log-pow-rev N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. unpow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

9. Applied rewrites 97.5%

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

    1. lift-*.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-log.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. lift--.f32 N/A

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

    7. associate-*l* N/A

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

    8. pow2 N/A

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

    9. negate-sub N/A

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

    10. mul-1-neg N/A

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

    11. lower-*.f32 N/A

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

    12. lift-log.f32 N/A

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

    13. lower-*.f32 N/A

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

    14. lift-PI.f32 N/A

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

    15. mul-1-neg N/A

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

    16. negate-sub N/A

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

    17. pow2 N/A

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

    18. lift--.f32 N/A

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

    19. lift-*.f32 97.5

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

11. Applied rewrites 97.5%

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

Alternative 8: 95.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. 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. negate-sub N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   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

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

   7. pow-pow N/A

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

   8. log-pow-rev N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-log.f32 N/A

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

   15. lift-PI.f32 95.4

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

4. Applied rewrites 95.4%

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

Alternative 9: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) / (cosTheta * ((single(pi) * 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. Taylor expanded in alpha around inf

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

   1. lower-/.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. log-rec N/A

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

   9. lower-neg.f32 N/A

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

   10. lower-log.f32 1.8

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

4. Applied rewrites 1.8%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. frac-2neg N/A

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

   13. lower-/.f32 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

6. Applied rewrites 1.8%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 1.8

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

8. Applied rewrites 1.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-log.f32 1.8

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

10. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025110 
(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)))))