GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 10.1s

Alternatives: 9

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

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (log (pow alpha (* 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(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(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 98.7%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.7%

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

3. Applied egg-rr 98.7%

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

   1. unpow-prod-down 98.7%

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

5. Applied egg-rr 98.7%

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

   1. pow-sqr 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

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: 97.5% 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 98.7%

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

2. Taylor expanded in alpha around 0 98.0%

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

   1. mul-1-neg 98.0%

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

4. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 4: 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 98.7%

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

   1. associate-/r* 98.7%

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

   2. difference-of-sqr-1 98.2%

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

   3. *-commutative 98.2%

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

   4. times-frac 98.1%

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

   5. *-commutative 98.1%

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

   6. times-frac 98.2%

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

   7. difference-of-sqr-1 98.7%

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

   8. associate-/l/ 98.3%

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

   9. log-prod 98.2%

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

   10. count-2 98.2%

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

   11. *-commutative 98.2%

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

   12. fma-neg 98.3%

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

   13. metadata-eval 98.3%

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

   14. +-commutative 98.3%

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

3. Simplified 98.3%

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

   1. associate-/l/ 98.8%

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

   2. fma-udef 98.7%

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

   3. difference-of-sqr--1 98.3%

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

   4. times-frac 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

6. Taylor expanded in cosTheta around 0 96.0%

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

7. Final simplification 96.0%

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

Alternative 5: 66.8% accurate, 1.1× speedup

Math

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

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(-0.5) / Float32(Float32(pi) * Float32(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 98.7%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 68.6%

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

   1. associate-/r* 68.6%

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

   2. *-commutative 68.6%

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

   3. associate-/r* 68.6%

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

   4. associate-/r* 68.6%

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

   5. associate-/r* 68.6%

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

   6. unpow2 68.6%

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

   7. neg-mul-1 68.6%

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

   8. sub-neg 68.6%

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

   9. unpow2 68.6%

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

   10. *-commutative 68.6%

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

   11. associate-*r* 68.5%

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

   12. unpow2 68.5%

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

6. Simplified 68.5%

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

7. Final simplification 68.5%

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

Alternative 6: 66.8% accurate, 1.1× speedup

Math

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

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(-0.5) / Float32(Float32(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 98.7%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.7%

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

3. Applied egg-rr 98.7%

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

   1. unpow-prod-down 98.7%

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

5. Applied egg-rr 98.7%

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

   1. pow-sqr 98.8%

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

7. Simplified 98.8%

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

   1. difference-of-sqr-1 98.5%

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

9. Applied egg-rr 98.5%

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

10. Taylor expanded in alpha around 0 68.6%

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

    1. *-commutative 68.6%

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

    2. associate-*l* 68.6%

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

    3. *-commutative 68.6%

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

    4. neg-mul-1 68.6%

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

    5. unsub-neg 68.6%

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

    6. unpow2 68.6%

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

12. Simplified 68.6%

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

13. Final simplification 68.6%

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

Alternative 7: 66.9% accurate, 1.1× speedup

Math

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

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(Float32(-0.5) / 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. Taylor expanded in alpha around 0 68.6%

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

   1. associate-/r* 68.6%

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

   2. *-commutative 68.6%

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

   3. mul-1-neg 68.6%

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

   4. unsub-neg 68.6%

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

   5. unpow2 68.6%

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

4. Simplified 68.6%

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

5. Final simplification 68.6%

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

Alternative 8: 65.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) / (single(pi) * log((single(1.0) / 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. fma-neg 98.6%

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

   2. metadata-eval 98.6%

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

   3. *-lft-identity 98.6%

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

   4. *-lft-identity 98.6%

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

   5. log-prod 98.7%

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

   6. count-2 98.7%

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

   7. *-commutative 98.7%

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

   8. +-commutative 98.7%

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

   9. associate-*l* 98.7%

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

   10. fma-def 98.7%

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

   11. fma-neg 98.7%

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

   12. metadata-eval 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in alpha around 0 98.1%

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

   1. associate-*r* 98.1%

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

   2. *-commutative 98.1%

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

   3. mul-1-neg 98.1%

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

   4. unsub-neg 98.1%

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

   5. unpow2 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 96.5%

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

8. Taylor expanded in alpha around 0 67.4%

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

9. Taylor expanded in alpha around inf 67.4%

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

10. Final simplification 67.4%

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

Alternative 9: 65.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. fma-neg 98.6%

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

   2. metadata-eval 98.6%

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

   3. *-lft-identity 98.6%

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

   4. *-lft-identity 98.6%

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

   5. log-prod 98.7%

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

   6. count-2 98.7%

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

   7. *-commutative 98.7%

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

   8. +-commutative 98.7%

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

   9. associate-*l* 98.7%

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

   10. fma-def 98.7%

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

   11. fma-neg 98.7%

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

   12. metadata-eval 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in alpha around 0 98.1%

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

   1. associate-*r* 98.1%

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

   2. *-commutative 98.1%

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

   3. mul-1-neg 98.1%

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

   4. unsub-neg 98.1%

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

   5. unpow2 98.1%

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

6. Simplified 98.1%

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

7. Taylor expanded in cosTheta around 0 96.5%

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

8. Taylor expanded in alpha around 0 67.4%

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

9. Final simplification 67.4%

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

Reproduce

herbie shell --seed 2023213 
(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)))))