GTR1 distribution

Percentage Accurate: 98.6% → 98.7%

Time: 5.3s

Alternatives: 11

Speedup: 0.7×

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: 98.6% 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 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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: 98.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (*
  (/ (fma alpha alpha -1.0) (* (* (log alpha) 2.0) PI))
  (/ 1.0 (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) * 2.0f) * ((float) M_PI))) * (1.0f / 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(2.0)) * Float32(pi))) * Float32(Float32(1.0) / fma(Float32(cosTheta * cosTheta), fma(alpha, alpha, Float32(-1.0)), Float32(1.0))))
end

Derivation

1. Initial program 98.6%

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

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\log \alpha \cdot 2\right) \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}} \]
8. 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 98.6%

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

3. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\log \alpha \cdot \pi\right) \cdot 2}}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)}} \]
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 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.5

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

   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: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

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

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

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

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

8. Applied rewrites 97.7%

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

Alternative 6: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. unpow-prod-down N/A

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

   13. mul-1-neg N/A

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

   14. unpow-neg-2 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 97.7

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

4. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 97.7

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

6. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. sub-flip N/A

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

   15. mul-1-neg N/A

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

   16. lower-*.f32 N/A

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

8. Applied rewrites 97.7%

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

Alternative 7: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. unpow-prod-down N/A

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

   13. mul-1-neg N/A

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

   14. unpow-neg-2 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 97.7

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

4. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 97.7

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

6. Applied rewrites 97.7%

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

7. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 97.7

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

9. Applied rewrites 97.7%

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

Alternative 8: 95.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift--.f32 N/A

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

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

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

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

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

   6. lift-fma.f32 N/A

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

   7. mult-flip N/A

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

   8. lift-fma.f32 N/A

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

   9. pow2 N/A

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

   10. add-flip N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

8. Applied rewrites 95.4%

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

Alternative 9: 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 98.6%

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

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. 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 95.6

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

4. Applied rewrites 95.6%

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

Alternative 10: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   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. Add Preprocessing

Alternative 11: 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 98.6%

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

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-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   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 2025128 
(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)))))