GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 4.6s

Alternatives: 13

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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha - 1}{2 \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right) \cdot 0.5\right) \cdot \pi\right)}\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (- (* alpha alpha) 1.0)
  (*
   2.0
   (log
    (pow
     (* alpha alpha)
     (* (* (fma (* (fma alpha alpha -1.0) cosTheta) cosTheta 1.0) 0.5) PI))))))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / (2.0f * logf(powf((alpha * alpha), ((fmaf((fmaf(alpha, alpha, -1.0f) * cosTheta), cosTheta, 1.0f) * 0.5f) * ((float) M_PI)))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(2.0) * log((Float32(alpha * alpha) ^ Float32(Float32(fma(Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), cosTheta, Float32(1.0)) * Float32(0.5)) * 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-log.f32 N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-*.f32 98.6

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

3. Applied rewrites 98.6%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. lower-unsound-*.f32 N/A

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

   4. lower-unsound-pow.f32 N/A

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

   5. lower-unsound-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-unsound-/.f32 98.6

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-unsound-log.f32 98.6

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

   7. lift-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

7. Applied rewrites 98.6%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{2 \cdot \log \left({\left(\alpha \cdot \alpha\right)}^{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right) \cdot 0.5\right) \cdot \pi\right)}\right)}} \]
8. Add Preprocessing

Alternative 2: 98.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-log.f32 N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-*.f32 98.6

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

3. Applied rewrites 98.6%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(fma(Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), cosTheta, Float32(1.0)) * Float32(Float32(2.0) * log(alpha)))) / 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

3. Applied rewrites 98.4%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-unsound-log.f32 98.5

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

5. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(fma(Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), cosTheta, Float32(1.0)) * log(Float32(alpha * alpha))) * 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

3. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. remove-double-neg N/A

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

   8. lift--.f32 N/A

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

   9. sub-flip N/A

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

   10. distribute-neg-in N/A

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

   11. remove-double-neg N/A

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

   12. distribute-neg-in N/A

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

   13. lift-*.f32 N/A

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

   14. sqr-abs-rev N/A

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

   15. distribute-rgt-neg-in N/A

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

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

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

   17. sqr-neg-rev N/A

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

   18. sqr-abs-rev N/A

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

   19. lower-fma.f32 N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f32 98.5

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

3. Applied rewrites 98.5%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. remove-double-neg N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-neg-in N/A

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

   5. remove-double-neg N/A

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

   6. distribute-neg-in N/A

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

   7. lift-*.f32 N/A

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

   8. sqr-abs-rev N/A

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

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

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

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

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

   11. sqr-neg-rev N/A

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

   12. sqr-abs-rev N/A

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

   13. lower-fma.f32 N/A

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

   14. metadata-eval 98.4

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*r* N/A

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

3. Applied rewrites 98.4%

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

Alternative 7: 97.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.5

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

4. Applied rewrites 97.5%

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

Alternative 8: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(fma(Float32(-cosTheta), cosTheta, Float32(1.0)) * log(Float32(alpha * alpha))) * 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.5

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

6. Applied rewrites 97.5%

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

8. Applied rewrites 97.5%

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

Alternative 9: 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 10: 95.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

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. Taylor expanded in alpha around inf

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

   1. lower-*.f32 N/A

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

   2. lower-log.f32 N/A

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

   3. lower-/.f32 95.0

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

7. Applied rewrites 95.0%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 95.0

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

9. Applied rewrites 95.2%

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

Alternative 11: 95.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

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

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

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

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

   7. lift-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 95.0

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*r* N/A

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

6. Applied rewrites 95.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. mult-flip-rev N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 95.1

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

8. Applied rewrites 95.1%

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

9. Taylor expanded in cosTheta around 0

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

    1. lower-PI.f32 95.1

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

11. Applied rewrites 95.1%

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

Alternative 12: 65.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

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. Taylor expanded in alpha around 0

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

7. Applied rewrites 65.4%

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

   1. lift-/.f32 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

9. Applied rewrites 65.4%

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

Alternative 13: 65.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

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

7. Applied rewrites 65.4%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow-rev N/A

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

   5. lift-log.f32 N/A

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

   6. lower-*.f32 65.4

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

9. Applied rewrites 65.4%

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

10. Taylor expanded in cosTheta around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 N/A

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

    4. lower-log.f32 65.4

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

12. Applied rewrites 65.4%

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

Reproduce

herbie shell --seed 2025157 
(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)))))