GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 3.9s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (/ (/ (fma alpha alpha -1.0) PI) (* (log alpha) 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) / ((float) M_PI)) / (logf(alpha) * 2.0f)) / fmaf((cosTheta * cosTheta), fmaf(alpha, alpha, -1.0f), 1.0f);
}

Julia

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

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. pow2 N/A

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

   8. add-flip N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. log-pow-rev N/A

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

   13. associate-/r* N/A

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

   14. pow2 N/A

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

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

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

   16. +-commutative N/A

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

   17. lower-/.f32 N/A

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

5. Applied rewrites 98.4%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. pow2 N/A

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

   8. add-flip N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. log-pow-rev N/A

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

   13. associate-/r* N/A

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

   14. pow2 N/A

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

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

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

   16. +-commutative N/A

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

   17. lower-/.f32 N/A

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.5%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (fma alpha alpha -1.0)
  (*
   (* (log alpha) 2.0)
   (* PI (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) * 2.0f) * (((float) M_PI) * fmaf((fmaf(alpha, alpha, -1.0f) * cosTheta), cosTheta, 1.0f)));
}

Julia

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

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. sub-flip 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. log-pow-rev N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. log-pow-rev N/A

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

   18. *-commutative N/A

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

   19. lower-*.f32 N/A

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

   20. lower-log.f32 N/A

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

   21. lift-PI.f32 98.5

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

   22. lift-+.f32 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. lift-fma.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. log-pow-rev N/A

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

   12. lower-*.f32 N/A

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

   13. log-pow-rev N/A

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

   14. *-commutative N/A

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

   15. lift-log.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lower-*.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lift-fma.f32 N/A

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

   20. pow2 N/A

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

5. Applied rewrites 98.5%

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

Alternative 4: 97.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. pow2 N/A

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

   8. add-flip N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. log-pow-rev N/A

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

   13. associate-/r* N/A

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

   14. pow2 N/A

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

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

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

   16. +-commutative N/A

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

   17. lower-/.f32 N/A

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

   4. distribute-lft-neg-out N/A

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

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lift-neg.f32 97.3

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

8. Applied rewrites 97.3%

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

Alternative 5: 97.3% accurate, 1.3× speedup

Math

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

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(pi)) * Float32(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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f32 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 97.4

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

4. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 97.4

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

   6. log-pow-rev 97.4

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-log.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. lift-*.f32 N/A

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

6. Applied rewrites 97.4%

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

Alternative 6: 95.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. pow2 N/A

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

   8. add-flip N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. log-pow-rev N/A

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

   13. associate-/r* N/A

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

   14. pow2 N/A

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

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

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

   16. +-commutative N/A

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

   17. lower-/.f32 N/A

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 95.1%

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

Alternative 7: 95.2% 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: 95.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. 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. sub-flip 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. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. log-pow-rev N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lift-PI.f32 95.2

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

4. Applied rewrites 95.2%

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

Alternative 9: 65.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

   1. associate--l+ N/A

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

   2. associate-*r/ N/A

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

   3. div-sub N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f32 N/A

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in alpha around 0

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

   1. sub-flip N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

   4. distribute-lft-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f32 66.5

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

7. Applied rewrites 66.5%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 65.9%

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f32 N/A

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

12. Applied rewrites 65.9%

    \[\leadsto \frac{\frac{-1}{\log \alpha \cdot 2}}{\color{blue}{\pi}} \]
13. Add Preprocessing

Alternative 10: 65.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

   1. associate--l+ N/A

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

   2. associate-*r/ N/A

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

   3. div-sub N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f32 N/A

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in alpha around 0

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

   1. sub-flip N/A

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

   2. mul-1-neg N/A

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

   3. pow2 N/A

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

   4. distribute-lft-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f32 66.5

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

7. Applied rewrites 66.5%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 65.9%

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

    1. lift-*.f32 N/A

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

    2. lift-log.f32 N/A

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

    3. *-commutative N/A

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

    4. log-pow-rev N/A

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

    5. lower-log.f32 N/A

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

    6. pow2 N/A

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

    7. lower-*.f32 65.9

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

12. Applied rewrites 65.9%

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

Reproduce

herbie shell --seed 2025132 
(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)))))