GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 3.2s

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: 98.5% 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: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 98.5

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

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

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

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

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

   15. lower-fma.f32 98.5

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

3. Applied rewrites 98.5%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

5. Applied rewrites 98.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 98.5

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

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

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

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

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

   15. lower-fma.f32 98.5

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

3. Applied rewrites 98.5%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-fma.f32 N/A

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

   10. lift-fma.f32 N/A

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

   11. associate-/l/ N/A

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

   12. lift-fma.f32 N/A

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

   13. pow2 N/A

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

   14. add-flip N/A

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

   15. metadata-eval N/A

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

5. Applied rewrites 98.6%

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

Alternative 5: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. count-2-rev N/A

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

   6. sum-log N/A

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

   7. pow2 N/A

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

   8. log-pow-rev N/A

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

   9. metadata-eval N/A

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

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

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

   11. distribute-rgt-neg-out N/A

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

   12. log-rec N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

4. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. lift-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. pow2 N/A

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

   9. add-flip N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

6. Applied rewrites 97.4%

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

Alternative 6: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. count-2-rev N/A

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

   6. sum-log N/A

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

   7. pow2 N/A

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

   8. log-pow-rev N/A

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

   9. metadata-eval N/A

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

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

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

   11. distribute-rgt-neg-out N/A

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

   12. log-rec N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

4. Applied rewrites 97.4%

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

Alternative 7: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. count-2-rev N/A

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

   6. sum-log N/A

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

   7. pow2 N/A

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

   8. log-pow-rev N/A

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

   9. metadata-eval N/A

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

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

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

   11. distribute-rgt-neg-out N/A

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

   12. log-rec N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

4. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 97.5

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

6. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-fma.f32 N/A

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

   8. pow2 N/A

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

   9. add-flip N/A

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

   18. metadata-eval N/A

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

   19. add-flip N/A

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

   20. pow2 N/A

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

   21. lift-fma.f32 N/A

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

   22. lift-log.f32 97.5

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

8. Applied rewrites 97.5%

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

Alternative 8: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. count-2-rev N/A

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

   6. sum-log N/A

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

   7. pow2 N/A

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

   8. log-pow-rev N/A

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

   9. metadata-eval N/A

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

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

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

   11. distribute-rgt-neg-out N/A

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

   12. log-rec N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

4. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 97.5

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

6. Applied rewrites 97.5%

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

Alternative 9: 95.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.1%

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

Alternative 10: 95.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

   1. log-pow-rev N/A

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

   2. pow2 N/A

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

   3. log-pow-rev N/A

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

   4. *-commutative N/A

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

   5. pow2 N/A

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

   6. log-pow-rev N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-log.f32 95.1

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

4. Applied rewrites 95.1%

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

Alternative 11: 95.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. sub-flip N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 N/A

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

   6. log-pow-rev N/A

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

   7. pow2 N/A

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

   8. log-pow-rev N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. log-pow-rev N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lift-PI.f32 N/A

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

   20. lower-log.f32 95.1

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

4. Applied rewrites 95.1%

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

Alternative 12: 66.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.1%

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

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 66.2%

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

8. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. log-pow-rev N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. lift-PI.f32 66.2

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

10. Applied rewrites 66.2%

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

Reproduce

herbie shell --seed 2025129 
(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)))))