GTR1 distribution

Percentage Accurate: 98.5% → 98.4%

Time: 12.9s

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: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. fma-neg 98.3%

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

   7. metadata-eval 98.3%

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

   8. *-commutative 98.3%

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

   9. distribute-rgt-neg-out 98.3%

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

   10. distribute-rgt-neg-out 98.3%

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

   11. distribute-lft-neg-in 98.3%

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

3. Simplified 98.3%

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

   1. clear-num 98.1%

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

   2. inv-pow 98.1%

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

   3. pow2 98.1%

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

   4. log-pow 98.2%

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

   5. associate-*r* 98.2%

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

5. Applied egg-rr 98.2%

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

   1. unpow-1 98.2%

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

   2. associate-/l* 98.5%

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

   3. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\log \left({\left({\alpha}^{2}\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ -1.0 (* alpha alpha))))
   (/
    t_0
    (* (log (pow (pow alpha 2.0) PI)) (+ 1.0 (* cosTheta (* cosTheta t_0)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = -1.0f + (alpha * alpha);
	return t_0 / (logf(powf(powf(alpha, 2.0f), ((float) M_PI))) * (1.0f + (cosTheta * (cosTheta * t_0))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(-1.0) + Float32(alpha * alpha))
	return Float32(t_0 / Float32(log(((alpha ^ Float32(2.0)) ^ Float32(pi))) * Float32(Float32(1.0) + Float32(cosTheta * Float32(cosTheta * t_0)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = single(-1.0) + (alpha * alpha);
	tmp = t_0 / (log(((alpha ^ single(2.0)) ^ single(pi))) * (single(1.0) + (cosTheta * (cosTheta * t_0))));
end

Derivation

1. Initial program 98.4%

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

   1. add-log-exp 98.4%

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

   2. *-commutative 98.4%

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

   3. exp-to-pow 98.5%

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

   4. pow2 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 3: 98.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. fma-neg 98.4%

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

   2. metadata-eval 98.4%

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

   3. *-commutative 98.4%

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

   4. fma-udef 98.4%

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

   5. distribute-rgt-in 98.4%

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

   6. pow2 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = single(-1.0) + (alpha * alpha);
	tmp = t_0 / ((single(1.0) + (cosTheta * (cosTheta * t_0))) * (single(pi) * log((alpha * alpha))));
end

Derivation

1. Initial program 98.4%

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

2. Final simplification 98.4%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in alpha around 0 97.7%

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

   1. mul-1-neg 97.7%

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

4. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 6: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \alpha}{2} \cdot \frac{\frac{\alpha + -1}{\pi}}{\log \alpha} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 94.8%

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

   1. metadata-eval 94.8%

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

   2. fma-neg 94.7%

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

   3. difference-of-sqr-1 94.7%

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

   4. *-un-lft-identity 94.7%

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

   5. fma-neg 94.7%

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

   6. metadata-eval 94.7%

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

   7. fma-def 94.7%

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

   8. *-un-lft-identity 94.7%

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

   9. *-commutative 94.7%

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

   10. *-commutative 94.7%

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

   11. frac-times 94.5%

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

   12. associate-*l/ 94.4%

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

   13. *-commutative 94.4%

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

   14. times-frac 94.5%

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

6. Applied egg-rr 94.5%

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

7. Final simplification 94.5%

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

Alternative 7: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \alpha}{\log \alpha} \cdot \frac{\frac{\alpha + -1}{\pi}}{2} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 94.8%

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

   1. metadata-eval 94.8%

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

   2. fma-neg 94.7%

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

   3. difference-of-sqr-1 94.7%

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

   4. *-un-lft-identity 94.7%

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

   5. fma-neg 94.7%

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

   6. metadata-eval 94.7%

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

   7. fma-def 94.7%

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

   8. *-un-lft-identity 94.7%

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

   9. *-commutative 94.7%

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

   10. *-commutative 94.7%

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

   11. frac-times 94.5%

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

   12. associate-*l/ 94.4%

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

   13. times-frac 94.5%

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

6. Applied egg-rr 94.5%

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

7. Final simplification 94.5%

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

Alternative 8: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \alpha}{2 \cdot \pi} \cdot \frac{\alpha + -1}{\log \alpha} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 94.8%

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

   1. metadata-eval 94.8%

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

   2. fma-neg 94.7%

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

   3. difference-of-sqr-1 94.7%

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

   4. *-un-lft-identity 94.7%

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

   5. fma-neg 94.7%

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

   6. metadata-eval 94.7%

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

   7. fma-def 94.7%

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

   8. *-un-lft-identity 94.7%

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

   9. associate-*r* 94.7%

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

   10. *-commutative 94.7%

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

   11. times-frac 94.6%

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

   12. *-commutative 94.6%

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

6. Applied egg-rr 94.6%

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

7. Final simplification 94.6%

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

Alternative 9: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 94.8%

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

   1. metadata-eval 94.8%

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

   2. fma-neg 94.7%

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

   3. difference-of-sqr-1 94.7%

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

   4. *-un-lft-identity 94.7%

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

   5. fma-neg 94.7%

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

   6. metadata-eval 94.7%

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

   7. fma-def 94.7%

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

   8. *-un-lft-identity 94.7%

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

   9. *-commutative 94.7%

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

   10. *-commutative 94.7%

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

   11. frac-times 94.5%

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

   12. associate-*l/ 94.4%

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

   13. times-frac 94.5%

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

6. Applied egg-rr 94.5%

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

   1. associate-*l/ 94.4%

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

   2. associate-*r/ 94.5%

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

   3. associate-/r* 94.5%

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

   4. associate-/r* 94.5%

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

   5. associate-*r/ 94.7%

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

   6. remove-double-neg 94.7%

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

   7. distribute-rgt-neg-out 94.7%

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

   8. *-commutative 94.7%

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

   9. distribute-rgt-neg-out 94.7%

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

   10. associate-/l* 94.6%

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

   11. neg-sub0 94.6%

       \[\leadsto \frac{\color{blue}{0 - \left(\alpha + -1\right)}}{\frac{\pi \cdot \left(2 \cdot \log \alpha\right)}{-\left(\alpha + 1\right)}} \]

   12. metadata-eval 94.6%

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

   13. +-commutative 94.6%

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

   14. associate--r+ 94.6%

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

   15. metadata-eval 94.6%

       \[\leadsto \frac{\left(\color{blue}{0} - -1\right) - \alpha}{\frac{\pi \cdot \left(2 \cdot \log \alpha\right)}{-\left(\alpha + 1\right)}} \]

   16. metadata-eval 94.6%

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

   17. associate-*r* 94.6%

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

   18. *-commutative 94.6%

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

   19. neg-sub0 94.6%

       \[\leadsto \frac{1 - \alpha}{\frac{\log \alpha \cdot \left(\pi \cdot 2\right)}{\color{blue}{0 - \left(\alpha + 1\right)}}} \]

8. Simplified 94.6%

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

9. Final simplification 94.6%

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

Alternative 10: 65.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. sqr-neg 98.4%

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

   8. sqr-neg 98.4%

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

   9. fma-neg 98.3%

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

   10. metadata-eval 98.3%

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

   11. associate-*l* 98.3%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta around 0 94.8%

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

   1. *-un-lft-identity 94.8%

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

   2. metadata-eval 94.8%

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

   3. times-frac 94.8%

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

   4. *-un-lft-identity 94.8%

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

   5. associate-*r* 94.8%

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

   6. *-commutative 94.8%

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

   7. frac-times 94.7%

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

   8. associate-*l/ 94.9%

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

   9. *-un-lft-identity 94.9%

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

   10. div-inv 94.8%

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

   11. *-commutative 94.8%

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

   12. times-frac 94.8%

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

6. Applied egg-rr 94.8%

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

7. Taylor expanded in alpha around 0 64.6%

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

8. Final simplification 64.6%

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

Alternative 11: 65.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in alpha around 0 66.5%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. mul-1-neg 66.5%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)\right)} \]

4. Simplified 66.5%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \left(-{cosTheta}^{2}\right)\right)\right)}} \]

5. Taylor expanded in cosTheta around 0 64.6%

   \[\leadsto \frac{-0.5}{\color{blue}{\pi \cdot \log \alpha}} \]
6. Step-by-step derivation

   1. clear-num 64.6%

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

   2. inv-pow 64.6%

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

7. Applied egg-rr 64.6%

   \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \log \alpha}{-0.5}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 64.6%

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

   2. associate-/l* 64.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\pi}{\frac{-0.5}{\log \alpha}}}} \]

9. Simplified 64.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{\frac{-0.5}{\log \alpha}}}} \]

10. Final simplification 64.6%

    \[\leadsto \frac{1}{\frac{\pi}{\frac{-0.5}{\log \alpha}}} \]

Alternative 12: 65.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in alpha around 0 66.5%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. mul-1-neg 66.5%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)\right)} \]

4. Simplified 66.5%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \left(-{cosTheta}^{2}\right)\right)\right)}} \]

5. Taylor expanded in cosTheta around 0 64.6%

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

6. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2023322 
(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)))))