GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 4.1s

Alternatives: 13

Speedup: 1.0×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

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

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

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

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-log.f32 N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-*.f32 98.7%

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

3. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.5%

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

3. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(fma(Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), cosTheta, Float32(1.0)) * Float32(pi)) * Float32(log(alpha) * Float32(2.0))))
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. remove-double-neg N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-neg-in N/A

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

   5. remove-double-neg N/A

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

   6. distribute-neg-in N/A

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

   7. lift-*.f32 N/A

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

   8. sqr-abs-rev N/A

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

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

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

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

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

   11. sqr-neg-rev N/A

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

   12. sqr-abs-rev N/A

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

   13. lower-fma.f32 N/A

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

   14. metadata-eval 98.4%

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*r* N/A

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

3. Applied rewrites 98.4%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow-rev N/A

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

   5. lift-log.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.5%

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

5. Applied rewrites 98.5%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. remove-double-neg N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. distribute-neg-in N/A

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

   5. remove-double-neg N/A

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

   6. distribute-neg-in N/A

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

   7. lift-*.f32 N/A

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

   8. sqr-abs-rev N/A

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

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

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

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

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

   11. sqr-neg-rev N/A

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

   12. sqr-abs-rev N/A

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

   13. lower-fma.f32 N/A

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

   14. metadata-eval 98.4%

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. associate-*r* N/A

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

3. Applied rewrites 98.4%

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

Alternative 5: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-*.f32 97.6%

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

4. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift--.f32 N/A

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

   5. sub-negate-rev N/A

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

   6. sub-negate-rev N/A

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

   7. lift--.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. lift-+.f32 N/A

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

   10. add-flip N/A

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

   11. sub-negate-rev N/A

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

6. Applied rewrites 97.6%

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

Alternative 6: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.6%

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

4. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.6%

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-fma.f32 97.6%

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

   11. lift-*.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 97.6%

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

6. Applied rewrites 97.6%

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

Alternative 7: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. lower-*.f32 97.6%

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

4. Applied rewrites 97.6%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 97.5%

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

6. Applied rewrites 97.5%

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

Alternative 8: 95.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. add-flip N/A

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

   8. metadata-eval N/A

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

   9. lower--.f32 95.0%

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

6. Applied rewrites 95.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift--.f32 N/A

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

   5. sub-flip N/A

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

   6. metadata-eval N/A

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

   7. lift--.f32 N/A

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

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

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

   9. lift-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

8. Applied rewrites 95.4%

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

Alternative 9: 95.4% accurate, 1.6× speedup

Math

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

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 95.4%

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

Alternative 10: 95.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(Float32(2.0) * Float32(log(alpha) * Float32(pi))) * 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 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 95.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. add-flip N/A

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

   8. metadata-eval N/A

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

   9. lower--.f32 95.0%

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

6. Applied rewrites 95.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-flip N/A

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

   5. metadata-eval N/A

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

   6. lift--.f32 N/A

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

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

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

   8. lift-fma.f32 95.3%

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-log.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. pow2 N/A

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

   14. log-pow-rev N/A

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

   15. lift-log.f32 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f32 N/A

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

   18. lower-*.f32 95.4%

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

8. Applied rewrites 95.4%

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

Alternative 11: 95.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in cosTheta around 0

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

   1. lower-PI.f32 95.3%

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

6. Applied rewrites 95.3%

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

Alternative 12: 65.7% accurate, 1.9× speedup

Math

\[\frac{\frac{-1}{1 \cdot \pi}}{\log \alpha \cdot 2} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) / (single(1.0) * single(pi))) / (log(alpha) * single(2.0));
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 95.4%

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

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 65.6%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

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

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

   6. lower-unsound-log.f32 65.6%

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

9. Applied rewrites 65.6%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lift-*.f32 N/A

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

    5. lift-log.f32 N/A

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

    6. log-pow-rev N/A

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

    7. pow2 N/A

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

    8. lift-*.f32 N/A

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

    9. lift-log.f32 N/A

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

    10. lift-*.f32 N/A

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

    11. associate-*l* N/A

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

    12. lift-*.f32 N/A

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

11. Applied rewrites 65.7%

    \[\leadsto \color{blue}{\frac{\frac{-1}{1 \cdot \pi}}{\log \alpha \cdot 2}} \]
12. Add Preprocessing

Alternative 13: 65.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.4%

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

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 65.6%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

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

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

   6. lower-unsound-log.f32 65.6%

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

9. Applied rewrites 65.6%

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

10. Taylor expanded in cosTheta around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 N/A

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

    4. lower-log.f32 65.6%

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

12. Applied rewrites 65.6%

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

Reproduce

herbie shell --seed 2025188 
(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)))))