GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 14.7s

Alternatives: 8

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, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

   1. add-log-exp 98.4%

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

   2. *-commutative 98.4%

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

   3. exp-to-pow 98.6%

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

   4. pow2 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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))) * log((alpha ^ Float32(Float32(2.0) * Float32(pi))))))
end

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. add-log-exp 98.4%

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

   2. *-commutative 98.4%

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

   3. exp-to-pow 98.6%

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

   4. pow2 98.6%

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

3. Applied egg-rr 98.6%

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

4. Taylor expanded in alpha around 0 98.3%

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

   1. associate-*r* 98.3%

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

   2. log-pow 98.5%

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

   3. rem-exp-log 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

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 98.4%

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

2. Final simplification 98.4%

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

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 98.4%

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

2. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

4. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 5: 95.2% 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 98.4%

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

   1. associate-/r* 98.3%

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

   2. cancel-sign-sub 98.3%

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

   3. distribute-rgt-neg-out 98.3%

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

   4. unsub-neg 98.3%

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

   5. distribute-rgt-neg-out 98.3%

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

   6. fma-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. *-commutative 98.2%

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

   9. distribute-rgt-neg-out 98.2%

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

   10. distribute-rgt-neg-out 98.2%

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

   11. distribute-lft-neg-in 98.2%

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

3. Simplified 98.2%

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

   1. div-inv 98.0%

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

   2. fma-udef 98.2%

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

   3. difference-of-sqr--1 97.9%

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

   4. add-exp-log 97.9%

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

   5. expm1-udef 97.9%

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

   6. associate-*l* 98.0%

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

   7. expm1-udef 98.0%

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

   8. add-exp-log 98.0%

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

   9. sub-neg 98.0%

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

   10. metadata-eval 98.0%

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

   11. pow2 98.0%

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

   12. log-pow 98.0%

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

   13. associate-*r* 98.0%

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

5. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

   3. associate-*r/ 98.0%

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

   4. times-frac 98.0%

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

   5. *-commutative 98.0%

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

7. Simplified 98.0%

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

   1. clear-num 97.8%

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

   2. frac-times 98.0%

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

   3. *-un-lft-identity 98.0%

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

9. Applied egg-rr 98.0%

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

10. Taylor expanded in alpha around 0 96.9%

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

    1. mul-1-neg 96.9%

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

12. Simplified 96.9%

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

13. Taylor expanded in cosTheta around 0 95.7%

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

14. Final simplification 95.7%

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

Alternative 6: 66.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-/r* 98.3%

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

   2. cancel-sign-sub 98.3%

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

   3. distribute-rgt-neg-out 98.3%

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

   4. unsub-neg 98.3%

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

   5. distribute-rgt-neg-out 98.3%

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

   6. fma-neg 98.2%

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

   7. metadata-eval 98.2%

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

   8. *-commutative 98.2%

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

   9. distribute-rgt-neg-out 98.2%

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

   10. distribute-rgt-neg-out 98.2%

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

   11. distribute-lft-neg-in 98.2%

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

3. Simplified 98.2%

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

   1. div-inv 98.0%

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

   2. fma-udef 98.2%

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

   3. difference-of-sqr--1 97.9%

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

   4. add-exp-log 97.9%

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

   5. expm1-udef 97.9%

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

   6. associate-*l* 98.0%

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

   7. expm1-udef 98.0%

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

   8. add-exp-log 98.0%

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

   9. sub-neg 98.0%

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

   10. metadata-eval 98.0%

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

   11. pow2 98.0%

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

   12. log-pow 98.0%

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

   13. associate-*r* 98.0%

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

5. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

   3. associate-*r/ 98.0%

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

   4. times-frac 98.0%

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

   5. *-commutative 98.0%

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

7. Simplified 98.0%

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

   1. clear-num 97.8%

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

   2. frac-times 98.0%

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

   3. *-un-lft-identity 98.0%

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

9. Applied egg-rr 98.0%

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

10. Taylor expanded in alpha around 0 96.9%

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

    1. mul-1-neg 96.9%

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

12. Simplified 96.9%

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

13. Taylor expanded in alpha around 0 64.5%

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

    1. associate-/r* 64.5%

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

15. Simplified 64.5%

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

16. Final simplification 64.5%

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

Alternative 7: 65.4% accurate, 1.1× speedup

Math

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

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(0.5) / Float32(log(alpha) * Float32(-Float32(pi))))
end

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

4. Simplified 97.3%

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

5. Taylor expanded in alpha around 0 64.5%

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

6. Taylor expanded in cosTheta around 0 64.0%

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

   1. log-pow 64.0%

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

   2. rem-exp-log -0.0%

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

   3. associate-*r* -0.0%

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

   4. *-commutative -0.0%

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

   5. log-prod -0.0%

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

   6. +-commutative -0.0%

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

   7. exp-sum -0.0%

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

   8. rem-exp-log 63.8%

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

   9. rem-exp-log 64.0%

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

8. Simplified 64.0%

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

9. Taylor expanded in alpha around inf 64.1%

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

    1. log-rec 64.0%

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

11. Simplified 64.0%

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

12. Final simplification 64.0%

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

Alternative 8: 65.5% accurate, 1.1× speedup

Math

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

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(Float32(0.5) / Float32(pi)) / Float32(-log(alpha)))
end

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

4. Simplified 97.3%

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

5. Taylor expanded in alpha around 0 64.5%

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

6. Taylor expanded in cosTheta around 0 64.0%

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

   1. log-pow 64.0%

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

   2. rem-exp-log -0.0%

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

   3. associate-*r* -0.0%

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

   4. *-commutative -0.0%

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

   5. log-prod -0.0%

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

   6. +-commutative -0.0%

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

   7. exp-sum -0.0%

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

   8. rem-exp-log 63.8%

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

   9. rem-exp-log 64.0%

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

8. Simplified 64.0%

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

9. Taylor expanded in alpha around inf 64.1%

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

    1. associate-/r* 64.1%

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

    2. log-rec 64.1%

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

11. Simplified 64.1%

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

12. Final simplification 64.1%

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

Reproduce

herbie shell --seed 2023339 
(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)))))