GTR1 distribution

Percentage Accurate: 98.5% → 98.4%

Time: 4.5s

Alternatives: 10

Speedup: N/A×

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: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in alpha around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 98.4

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

6. Applied rewrites 98.4%

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

Alternative 2: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

3. Applied rewrites 98.4%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

3. Applied rewrites 98.4%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-fma.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-/l/ N/A

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

5. Applied rewrites 98.4%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(pi) * Float32(log(Float32(alpha * alpha)) * fma(fma(Float32(alpha * alpha), cosTheta, 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. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 98.5

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 98.5

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

6. Applied rewrites 98.5%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 98.5

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 98.5

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f32 97.5

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

9. Applied rewrites 97.5%

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-/r* N/A

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

    5. lift--.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. lift-*.f32 N/A

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

    9. lift--.f32 N/A

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

    10. lift-/.f32 N/A

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

    11. lift-PI.f32 97.5

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

11. Applied rewrites 97.5%

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

Alternative 6: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 98.5

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 98.5

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f32 97.5

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

9. Applied rewrites 97.5%

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

Alternative 7: 95.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 95.1

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

4. Applied rewrites 95.1%

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

Alternative 8: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / (single(pi) * log((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 alpha around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 98.5

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 98.5

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in cosTheta around 0

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

   1. lower-log.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 95.2

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

9. Applied rewrites 95.2%

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

Alternative 9: 95.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{1}} \]
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 65.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. log-pow-rev N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

   8. log-pow-rev N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-log.f32 N/A

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

   15. lift-PI.f32 65.3

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

9. Applied rewrites 65.3%

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

10. Taylor expanded in cosTheta around 0

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

12. Applied rewrites 95.1%

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

Alternative 10: 65.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = 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. 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 65.3%

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

Reproduce

herbie shell --seed 2025107 
(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)))))