GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 19.0s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. add-log-exp 98.7%

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

   2. *-commutative 98.7%

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

   3. exp-to-pow 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Final simplification 98.7%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-/r* 98.8%

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

   2. associate-/r* 98.5%

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

   3. log-prod 98.4%

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

   4. count-2 98.4%

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

   5. fma-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. *-commutative 98.4%

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

   8. +-commutative 98.4%

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

   9. associate-*l* 98.4%

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

   10. fma-def 98.4%

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

   11. fma-neg 98.4%

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

   12. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in alpha around 0 96.8%

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

   1. clear-num 96.7%

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

   2. inv-pow 96.8%

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

6. Applied egg-rr 96.8%

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

8. Simplified 96.7%

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

   1. div-inv 96.7%

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

   2. metadata-eval 96.7%

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

   3. fma-neg 96.7%

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

   4. associate-/r/ 96.5%

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

   5. fma-neg 96.6%

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

   6. metadata-eval 96.6%

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

   7. mul-1-neg 96.6%

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

   8. unpow2 96.6%

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

10. Applied egg-rr 96.6%

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

    1. un-div-inv 96.6%

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

    2. associate-*l/ 96.8%

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

    3. *-un-lft-identity 96.8%

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

    4. associate-/l/ 97.1%

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

    5. *-commutative 97.1%

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

    6. distribute-lft-neg-in 97.1%

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

    7. associate-/r* 97.1%

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

    8. frac-2neg 97.1%

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

    9. associate-*l* 97.1%

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

    10. distribute-lft-neg-in 97.1%

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

    11. +-commutative 97.1%

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

    12. distribute-rgt-neg-in 97.1%

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

    13. fma-def 97.1%

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

12. Applied egg-rr 97.1%

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

    1. neg-sub0 97.1%

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

    2. metadata-eval 97.1%

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

    3. fma-def 97.1%

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

    4. unpow2 97.1%

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

    5. +-commutative 97.1%

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

    6. associate--r+ 97.1%

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

    7. metadata-eval 97.1%

       \[\leadsto \frac{\left(\color{blue}{0} - -1\right) - {\alpha}^{2}}{-\pi \cdot \left(\left(\log \alpha \cdot 2\right) \cdot \mathsf{fma}\left(cosTheta, -cosTheta, 1\right)\right)} \]

    8. metadata-eval 97.1%

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

    9. unpow2 97.1%

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

    10. distribute-rgt-neg-in 97.1%

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

    11. *-commutative 97.1%

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

    12. fma-udef 97.1%

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

    13. distribute-rgt-neg-out 97.1%

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

    14. unpow2 97.1%

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

    15. mul-1-neg 97.1%

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

    16. +-commutative 97.1%

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

    17. distribute-rgt-neg-in 97.1%

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

    18. mul-1-neg 97.1%

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

    19. sub-neg 97.1%

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

    20. unpow2 97.1%

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

14. Simplified 97.1%

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

15. Final simplification 97.1%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in alpha around 0 97.1%

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

   1. associate-*r* 97.1%

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

   2. mul-1-neg 97.1%

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

   3. sub-neg 97.1%

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

   4. unpow2 97.1%

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

9. Simplified 97.1%

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

10. Final simplification 97.1%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 97.1%

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

6. Simplified 97.1%

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

   1. add-log-exp 97.1%

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

   2. distribute-lft-neg-in 97.1%

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

   3. exp-prod 97.1%

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

   4. *-commutative 97.1%

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

   5. *-commutative 97.1%

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

   6. log-pow 97.2%

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

   7. pow2 97.2%

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

   8. exp-to-pow 97.3%

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

   9. +-commutative 97.3%

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

   10. distribute-rgt-neg-in 97.3%

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

   11. fma-def 97.3%

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

8. Applied egg-rr 97.3%

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

   1. log-pow 97.2%

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

   2. fma-udef 97.3%

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

   3. distribute-rgt-neg-out 97.3%

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

   4. unpow2 97.3%

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

   5. mul-1-neg 97.3%

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

   6. +-commutative 97.3%

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

   7. mul-1-neg 97.3%

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

   8. sub-neg 97.3%

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

   9. unpow2 97.3%

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

   10. unpow2 97.3%

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

   11. log-pow 97.2%

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

   12. unpow2 97.2%

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

10. Simplified 97.2%

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

11. Final simplification 97.2%

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

Alternative 6: 67.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. add-log-exp 98.7%

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

   2. *-commutative 98.7%

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

   3. exp-to-pow 98.8%

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

3. Applied egg-rr 98.8%

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

4. Taylor expanded in alpha around 0 66.9%

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

   1. *-commutative 66.9%

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

   2. neg-mul-1 66.9%

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

   3. unsub-neg 66.9%

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

   4. unpow2 66.9%

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

6. Simplified 66.9%

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

7. Final simplification 66.9%

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

Alternative 7: 67.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-/r* 98.8%

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

   2. associate-/r* 98.5%

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

   3. log-prod 98.4%

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

   4. count-2 98.4%

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

   5. fma-neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. *-commutative 98.4%

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

   8. +-commutative 98.4%

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

   9. associate-*l* 98.4%

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

   10. fma-def 98.4%

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

   11. fma-neg 98.4%

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

   12. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. fma-udef 98.4%

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

5. Applied egg-rr 98.4%

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

6. Taylor expanded in alpha around 0 66.9%

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

   1. mul-1-neg 66.9%

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

   2. sub-neg 66.9%

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

   3. *-commutative 66.9%

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

   4. sub-neg 66.9%

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

   5. mul-1-neg 66.9%

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

   6. associate-*r* 66.9%

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

   7. mul-1-neg 66.9%

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

   8. sub-neg 66.9%

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

   9. *-commutative 66.9%

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

   10. unpow2 66.9%

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

8. Simplified 66.9%

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

9. Final simplification 66.9%

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

Alternative 8: 95.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. pow2 98.7%

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

   2. log-pow 98.7%

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

   3. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in cosTheta around 0 93.5%

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

8. Final simplification 93.5%

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

Alternative 9: 65.8% 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 98.7%

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

2. Taylor expanded in cosTheta around 0 93.6%

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

4. Simplified 93.5%

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

5. Taylor expanded in alpha around 0 64.4%

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

   1. clear-num 64.4%

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

   2. inv-pow 64.4%

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

   3. *-commutative 64.4%

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

7. Applied egg-rr 64.4%

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

   1. unpow-1 64.4%

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

   2. associate-/l* 64.5%

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

9. Simplified 64.5%

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

10. Final simplification 64.5%

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

Alternative 10: 65.8% 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 98.7%

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

2. Taylor expanded in cosTheta around 0 93.6%

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

4. Simplified 93.5%

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

5. Taylor expanded in alpha around 0 64.4%

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

6. Final simplification 64.4%

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

Alternative 11: 65.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in cosTheta around 0 93.6%

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

4. Simplified 93.5%

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

5. Taylor expanded in alpha around 0 64.4%

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

   1. associate-/r* 64.5%

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

7. Simplified 64.5%

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

8. Final simplification 64.5%

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

Reproduce

herbie shell --seed 2023279 
(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)))))