GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 13.4s

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: 98.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (log (pow alpha (* PI 2.0))) (+ 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, (((float) M_PI) * 2.0f))) * (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((alpha ^ Float32(Float32(pi) * Float32(2.0)))) * 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 ^ (single(pi) * single(2.0)))) * (single(1.0) + (cosTheta * (t_0 * cosTheta))));
end

Derivation

1. Initial program 98.6%

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

   1. pow2 98.6%

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

   2. log-pow 98.5%

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

   3. associate-*l* 98.5%

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

   4. add-log-exp 98.5%

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

   5. *-commutative 98.5%

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

   6. exp-to-pow 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) + -1.0f;
	return t_0 / ((1.0f + (cosTheta * (t_0 * cosTheta))) * (2.0f * (((float) M_PI) * logf(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(2.0) * Float32(Float32(pi) * log(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(2.0) * (single(pi) * log(alpha))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 98.5%

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

3. Final simplification 98.5%

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

Alternative 3: 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 98.6%

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

2. Final simplification 98.6%

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

Alternative 4: 97.5% 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 98.6%

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

2. Taylor expanded in alpha around 0 97.5%

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

   1. mul-1-neg 97.5%

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

4. Simplified 97.5%

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

5. Final simplification 97.5%

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

Alternative 5: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* 98.6%

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

   2. fma-neg 98.4%

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

   3. metadata-eval 98.4%

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

   4. log-prod 98.5%

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

   5. distribute-rgt-in 98.5%

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

   6. distribute-lft-out 98.5%

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

   7. *-rgt-identity 98.5%

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

   8. *-rgt-identity 98.5%

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

   9. distribute-lft-out 98.5%

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

   10. metadata-eval 98.5%

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

   11. +-commutative 98.5%

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

   12. associate-*l* 98.5%

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

   13. fma-def 98.5%

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

   14. fma-neg 98.5%

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

   15. metadata-eval 98.5%

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

3. Simplified 98.5%

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

   1. metadata-eval 98.5%

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

   2. fma-neg 98.5%

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

   3. difference-of-sqr-1 98.3%

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

   4. add-exp-log 98.2%

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

   5. expm1-udef 98.3%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \alpha\right)}}{\log \alpha \cdot \left(\pi \cdot 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   6. associate-*r* 98.3%

      \[\leadsto \frac{\frac{\left(\alpha + 1\right) \cdot \mathsf{expm1}\left(\log \alpha\right)}{\color{blue}{\left(\log \alpha \cdot \pi\right) \cdot 2}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   7. times-frac 98.3%

      \[\leadsto \frac{\color{blue}{\frac{\alpha + 1}{\log \alpha \cdot \pi} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{2}}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   8. *-commutative 98.3%

      \[\leadsto \frac{\frac{\alpha + 1}{\color{blue}{\pi \cdot \log \alpha}} \cdot \frac{\mathsf{expm1}\left(\log \alpha\right)}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

   9. expm1-udef 98.3%

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

   10. add-exp-log 98.3%

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

   11. sub-neg 98.3%

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

   12. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in cosTheta around 0 95.1%

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

7. Final simplification 95.1%

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

Alternative 6: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 97.5%

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

   1. mul-1-neg 97.5%

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

4. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 95.5%

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

   1. unpow2 95.5%

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

   2. fma-neg 95.3%

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

   3. metadata-eval 95.3%

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

   4. log-pow 95.5%

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

   5. associate-*l* 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 95.5%

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

   8. log-pow 95.4%

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

7. Simplified 95.4%

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

   1. metadata-eval 95.4%

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

   2. fma-neg 95.4%

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

   3. log-pow 95.5%

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

   4. pow-unpow 95.5%

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

   5. difference-of-sqr-1 95.3%

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

   6. *-un-lft-identity 95.3%

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

   7. fma-neg 95.3%

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

   8. metadata-eval 95.3%

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

   9. fma-def 95.3%

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

   10. *-un-lft-identity 95.3%

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

   11. log-pow 95.1%

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

   12. frac-times 95.1%

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

   13. associate-*l/ 95.1%

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

   14. *-commutative 95.1%

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

   15. times-frac 95.1%

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

9. Applied egg-rr 95.1%

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

    1. associate-*r/ 95.1%

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

    2. div-inv 95.1%

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

    3. metadata-eval 95.1%

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

11. Applied egg-rr 95.1%

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

    1. associate-/l* 94.9%

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

    2. *-commutative 94.9%

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

    3. associate-/r/ 95.0%

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

    4. times-frac 95.2%

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

13. Simplified 95.2%

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

14. Final simplification 95.2%

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

Alternative 7: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 97.5%

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

   1. mul-1-neg 97.5%

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

4. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 95.5%

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

   1. unpow2 95.5%

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

   2. fma-neg 95.3%

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

   3. metadata-eval 95.3%

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

   4. log-pow 95.5%

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

   5. associate-*l* 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 95.5%

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

   8. log-pow 95.4%

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

7. Simplified 95.4%

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

   1. metadata-eval 95.4%

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

   2. fma-neg 95.4%

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

   3. difference-of-sqr-1 95.2%

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

   4. *-un-lft-identity 95.2%

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

   5. fma-neg 95.2%

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

   6. metadata-eval 95.2%

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

   7. fma-def 95.2%

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

   8. *-un-lft-identity 95.2%

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

   9. *-commutative 95.2%

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

   10. log-pow 95.2%

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

   11. pow-unpow 95.3%

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

   12. log-pow 95.1%

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

   13. times-frac 95.2%

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

9. Applied egg-rr 95.2%

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

10. Final simplification 95.2%

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

Alternative 8: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in alpha around 0 97.5%

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

   1. mul-1-neg 97.5%

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

4. Simplified 97.5%

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

5. Taylor expanded in cosTheta around 0 95.5%

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

   1. unpow2 95.5%

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

   2. fma-neg 95.3%

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

   3. metadata-eval 95.3%

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

   4. log-pow 95.5%

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

   5. associate-*l* 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 95.5%

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

   8. log-pow 95.4%

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

7. Simplified 95.4%

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

   1. metadata-eval 95.4%

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

   2. fma-neg 95.4%

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

   3. difference-of-sqr-1 95.2%

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

   4. *-un-lft-identity 95.2%

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

   5. fma-neg 95.2%

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

   6. metadata-eval 95.2%

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

   7. fma-def 95.2%

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

   8. *-un-lft-identity 95.2%

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

   9. times-frac 95.1%

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

   10. log-pow 95.2%

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

9. Applied egg-rr 95.2%

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

10. Final simplification 95.2%

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

Alternative 9: 65.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in cosTheta around 0 95.5%

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

   1. unpow2 95.5%

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

   2. fma-neg 95.3%

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

   3. metadata-eval 95.3%

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

   4. *-lft-identity 95.3%

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

   5. log-pow 95.5%

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

   6. *-commutative 95.5%

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

   7. times-frac 95.1%

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

   8. associate-/r* 95.1%

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

   9. times-frac 95.4%

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

   10. *-commutative 95.4%

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

   11. times-frac 95.4%

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

   12. metadata-eval 95.4%

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

4. Simplified 95.4%

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

5. Taylor expanded in alpha around 0 63.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{-1}{\pi \cdot \log \alpha}} \]
6. Step-by-step derivation

   1. associate-/r* 63.8%

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

7. Simplified 63.8%

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

   1. div-inv 63.8%

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

9. Applied egg-rr 63.8%

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

10. Final simplification 63.8%

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

Alternative 10: 65.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in cosTheta around 0 95.5%

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

   1. unpow2 95.5%

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

   2. fma-neg 95.3%

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

   3. metadata-eval 95.3%

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

   4. *-lft-identity 95.3%

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

   5. log-pow 95.5%

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

   6. *-commutative 95.5%

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

   7. times-frac 95.1%

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

   8. associate-/r* 95.1%

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

   9. times-frac 95.4%

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

   10. *-commutative 95.4%

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

   11. times-frac 95.4%

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

   12. metadata-eval 95.4%

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

4. Simplified 95.4%

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

5. Taylor expanded in alpha around 0 63.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{-1}{\pi \cdot \log \alpha}} \]
6. Step-by-step derivation

   1. associate-/r* 63.8%

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

7. Simplified 63.8%

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

8. Final simplification 63.8%

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

Alternative 11: 65.6% 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 98.6%

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

2. Taylor expanded in alpha around 0 64.8%

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

   1. mul-1-neg 64.8%

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

4. Simplified 64.8%

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

5. Taylor expanded in cosTheta around 0 63.8%

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

6. Final simplification 63.8%

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

Reproduce

herbie shell --seed 2023308 
(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)))))