GTR1 distribution

Percentage Accurate: 98.6% → 98.6%

Time: 3.8s

Alternatives: 12

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: 98.6% 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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(pi)) / Float32(fma(Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), cosTheta, Float32(1.0)) * log(Float32(alpha * 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. 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. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

3. Applied rewrites 98.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

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(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(Float32(fma(Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), cosTheta, Float32(1.0)) * Float32(2.0)) * log(alpha)) * Float32(pi)))
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

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

   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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

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

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

   4. metadata-eval N/A

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

   5. sub-flip N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.1

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

5. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. frac-times N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. metadata-eval N/A

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

   13. lift--.f32 N/A

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

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

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

   15. lift-fma.f32 N/A

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

7. Applied rewrites 98.6%

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

Alternative 3: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(pi)) / Float32(fma(Float32(Float32(-1.0) * cosTheta), cosTheta, Float32(1.0)) * log(Float32(alpha * 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. 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. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.5

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

6. Applied rewrites 97.5%

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

Alternative 4: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

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

   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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

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

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

   4. metadata-eval N/A

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

   5. sub-flip N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.1

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

5. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. frac-times N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. metadata-eval N/A

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

   13. lift--.f32 N/A

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

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

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

   15. lift-fma.f32 N/A

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.7

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

10. Applied rewrites 97.7%

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

Alternative 5: 97.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

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

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

   4. metadata-eval N/A

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

   5. sub-flip N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 97.2%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. lift--.f32 N/A

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

   7. sub-flip N/A

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

   8. metadata-eval N/A

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

   9. lift--.f32 N/A

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

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

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

   11. lift-fma.f32 N/A

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

10. Applied rewrites 97.7%

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

Alternative 6: 95.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(pi)) / Float32(Float32(1.0) * log(Float32(alpha * 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. 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. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.3%

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

Alternative 7: 95.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Applied rewrites 95.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
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. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

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

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

   5. metadata-eval N/A

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

   6. sub-flip N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. times-frac N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 N/A

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

6. Applied rewrites 95.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

8. Applied rewrites 95.4%

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

Alternative 8: 95.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(pi) * Float32(log(alpha) * 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. 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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

   1. lower-PI.f32 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 95.4

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

   7. lift-*.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. pow2 N/A

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

   10. log-pow-rev N/A

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

   11. lift-log.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 95.5

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

8. Applied rewrites 95.5%

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

Alternative 9: 95.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

   1. lower-PI.f32 95.4

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

6. Applied rewrites 95.4%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. mult-flip N/A

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

   5. lower-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

8. Applied rewrites 95.3%

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

Alternative 10: 65.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) / single(pi)) * -(single(-0.5) / 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. 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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

   1. lower-PI.f32 95.4

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 65.8%

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

    1. lift-/.f32 N/A

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

    2. mult-flip N/A

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

    3. lift-/.f32 N/A

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

    4. frac-times N/A

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

    5. *-commutative N/A

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

11. Applied rewrites 65.8%

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

Alternative 11: 65.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. lift-/.f32 N/A

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

   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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

   1. lower-PI.f32 95.4

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 65.8%

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

Alternative 12: 65.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

   1. lower-PI.f32 95.4

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

6. Applied rewrites 95.4%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 65.8%

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

10. Taylor expanded in alpha around 0

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

    1. lower-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{-1}{2}}{\color{blue}{\log \alpha}}}{\pi} \]

    2. lower-log.f32 65.8

       \[\leadsto \frac{\frac{-0.5}{\log \alpha}}{\pi} \]

12. Applied rewrites 65.8%

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

Reproduce

herbie shell --seed 2025156 
(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)))))