GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 1.6min

Alternatives: 8

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ t_1 := {\left(\alpha \cdot \alpha\right)}^{\pi}\\ \frac{t\_0}{\log \left({t\_1}^{1}\right) + \log \left({t\_1}^{\left(\left(cosTheta \cdot cosTheta\right) \cdot t\_0\right)}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)) (t_1 (pow (* alpha alpha) PI)))
   (/
    t_0
    (+ (log (pow t_1 1.0)) (log (pow t_1 (* (* cosTheta cosTheta) t_0)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	float t_1 = powf((alpha * alpha), ((float) M_PI));
	return t_0 / (logf(powf(t_1, 1.0f)) + logf(powf(t_1, ((cosTheta * cosTheta) * t_0))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	t_1 = Float32(alpha * alpha) ^ Float32(pi)
	return Float32(t_0 / Float32(log((t_1 ^ Float32(1.0))) + log((t_1 ^ Float32(Float32(cosTheta * cosTheta) * t_0)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) - single(1.0);
	t_1 = (alpha * alpha) ^ single(pi);
	tmp = t_0 / (log((t_1 ^ single(1.0))) + log((t_1 ^ ((cosTheta * cosTheta) * t_0))));
end

Derivation

1. Initial program 98.4%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-in N/A

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

   12. lower-+.f32 N/A

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

4. Applied rewrites 98.6%

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

Alternative 2: 98.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(\alpha \cdot \alpha\right)\\ \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(t\_0, t\_0, 1 + \alpha \cdot \alpha\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (* -1.0 (* alpha alpha))))
   (/
    (- (* alpha alpha) 1.0)
    (*
     (* PI (log (* alpha alpha)))
     (+
      1.0
      (*
       (*
        (/
         (- (pow (* alpha alpha) 3.0) 1.0)
         (fma t_0 t_0 (+ 1.0 (* alpha alpha))))
        cosTheta)
       cosTheta))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = -1.0f * (alpha * alpha);
	return ((alpha * alpha) - 1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((((powf((alpha * alpha), 3.0f) - 1.0f) / fmaf(t_0, t_0, (1.0f + (alpha * alpha)))) * cosTheta) * cosTheta)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(-1.0) * Float32(alpha * alpha))
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(Float32(Float32((Float32(alpha * alpha) ^ Float32(3.0)) - Float32(1.0)) / fma(t_0, t_0, Float32(Float32(1.0) + Float32(alpha * alpha)))) * 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. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. flip3-- N/A

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

   5. lower-/.f32 N/A

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

   6. pow-pow N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\alpha}^{\color{blue}{6}} - {1}^{3}}{{\alpha}^{2} \cdot {\alpha}^{2} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\alpha}^{6} - \color{blue}{1}}{{\alpha}^{2} \cdot {\alpha}^{2} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{\color{blue}{{\alpha}^{6} - 1}}{{\alpha}^{2} \cdot {\alpha}^{2} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. lower-pow.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 N/A

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

   15. sqr-neg-rev N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\color{blue}{\left(\mathsf{neg}\left({\alpha}^{2}\right)\right) \cdot \left(\mathsf{neg}\left({\alpha}^{2}\right)\right)} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   16. lower-fma.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left({\alpha}^{2}\right), \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)}} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   17. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(\color{blue}{-{\alpha}^{2}}, \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   18. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\color{blue}{\alpha \cdot \alpha}, \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   19. lift-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\color{blue}{\alpha \cdot \alpha}, \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   20. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, \color{blue}{-{\alpha}^{2}}, 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   21. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\color{blue}{\alpha \cdot \alpha}, 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   22. lift-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\color{blue}{\alpha \cdot \alpha}, 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   23. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\alpha \cdot \alpha, \color{blue}{1} + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\color{blue}{\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\alpha \cdot \alpha, 1 + \left(\alpha \cdot \alpha\right) \cdot 1\right)}} \cdot cosTheta\right) \cdot cosTheta\right)} \]

5. Final simplification 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-1 \cdot \left(\alpha \cdot \alpha\right), -1 \cdot \left(\alpha \cdot \alpha\right), 1 + \alpha \cdot \alpha\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]
6. Add Preprocessing

Alternative 3: 98.5% accurate, N/A× 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

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. Add Preprocessing
3. Add Preprocessing

Alternative 4: 98.3% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.1%

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

   1. lift--.f32 N/A

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

   2. flip3-- N/A

      \[\leadsto \frac{\alpha + 1}{\log \left({\left({\alpha}^{\pi}\right)}^{2}\right)} \cdot \frac{\color{blue}{\frac{{\alpha}^{3} - {1}^{3}}{\alpha \cdot \alpha + \left(1 \cdot 1 + \alpha \cdot 1\right)}}}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \alpha \cdot \alpha - 1, 1\right)} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{\alpha + 1}{\log \left({\left({\alpha}^{\pi}\right)}^{2}\right)} \cdot \frac{\color{blue}{\frac{{\alpha}^{3} - {1}^{3}}{\alpha \cdot \alpha + \left(1 \cdot 1 + \alpha \cdot 1\right)}}}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \alpha \cdot \alpha - 1, 1\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\alpha + 1}{\log \left({\left({\alpha}^{\pi}\right)}^{2}\right)} \cdot \frac{\frac{{\alpha}^{3} - \color{blue}{1}}{\alpha \cdot \alpha + \left(1 \cdot 1 + \alpha \cdot 1\right)}}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \alpha \cdot \alpha - 1, 1\right)} \]

   5. lower--.f32 N/A

      \[\leadsto \frac{\alpha + 1}{\log \left({\left({\alpha}^{\pi}\right)}^{2}\right)} \cdot \frac{\frac{\color{blue}{{\alpha}^{3} - 1}}{\alpha \cdot \alpha + \left(1 \cdot 1 + \alpha \cdot 1\right)}}{\mathsf{fma}\left(cosTheta \cdot cosTheta, \alpha \cdot \alpha - 1, 1\right)} \]

   6. unpow3 N/A

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

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f32 N/A

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

   14. lower-*.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Final simplification 98.3%

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

Alternative 5: 98.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-1, cosTheta \cdot cosTheta, cosTheta \cdot cosTheta\right)\\ t_1 := 1 - cosTheta \cdot cosTheta\\ t_2 := \pi \cdot \left(\log \alpha \cdot {t\_1}^{2}\right)\\ t_3 := \frac{t\_0}{t\_2}\\ t_4 := \frac{1}{\pi \cdot \left(\log \alpha \cdot t\_1\right)}\\ t_5 := 0.5 \cdot t\_4\\ t_6 := -0.5 \cdot \frac{cosTheta \cdot cosTheta}{t\_2}\\ t_7 := t\_5 - t\_6\\ t_8 := \mathsf{fma}\left(0.5, t\_3, \frac{\left(cosTheta \cdot cosTheta\right) \cdot t\_7}{t\_1}\right)\\ \left(\alpha \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(0.5, t\_4, \left(\alpha \cdot \alpha\right) \cdot \left(\left(-1 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \mathsf{fma}\left(-1, \frac{t\_0 \cdot t\_7}{t\_1}, \mathsf{fma}\left(-1, \frac{\left(cosTheta \cdot cosTheta\right) \cdot t\_8}{t\_1}, -0.5 \cdot t\_3\right)\right) - t\_8\right)\right) - t\_6\right) - t\_5 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (fma -1.0 (* cosTheta cosTheta) (* cosTheta cosTheta)))
        (t_1 (- 1.0 (* cosTheta cosTheta)))
        (t_2 (* PI (* (log alpha) (pow t_1 2.0))))
        (t_3 (/ t_0 t_2))
        (t_4 (/ 1.0 (* PI (* (log alpha) t_1))))
        (t_5 (* 0.5 t_4))
        (t_6 (* -0.5 (/ (* cosTheta cosTheta) t_2)))
        (t_7 (- t_5 t_6))
        (t_8 (fma 0.5 t_3 (/ (* (* cosTheta cosTheta) t_7) t_1))))
   (-
    (*
     (* alpha alpha)
     (-
      (fma
       0.5
       t_4
       (*
        (* alpha alpha)
        (-
         (*
          (* -1.0 (* alpha alpha))
          (fma
           -1.0
           (/ (* t_0 t_7) t_1)
           (fma -1.0 (/ (* (* cosTheta cosTheta) t_8) t_1) (* -0.5 t_3))))
         t_8)))
      t_6))
    t_5)))

C

float code(float cosTheta, float alpha) {
	float t_0 = fmaf(-1.0f, (cosTheta * cosTheta), (cosTheta * cosTheta));
	float t_1 = 1.0f - (cosTheta * cosTheta);
	float t_2 = ((float) M_PI) * (logf(alpha) * powf(t_1, 2.0f));
	float t_3 = t_0 / t_2;
	float t_4 = 1.0f / (((float) M_PI) * (logf(alpha) * t_1));
	float t_5 = 0.5f * t_4;
	float t_6 = -0.5f * ((cosTheta * cosTheta) / t_2);
	float t_7 = t_5 - t_6;
	float t_8 = fmaf(0.5f, t_3, (((cosTheta * cosTheta) * t_7) / t_1));
	return ((alpha * alpha) * (fmaf(0.5f, t_4, ((alpha * alpha) * (((-1.0f * (alpha * alpha)) * fmaf(-1.0f, ((t_0 * t_7) / t_1), fmaf(-1.0f, (((cosTheta * cosTheta) * t_8) / t_1), (-0.5f * t_3)))) - t_8))) - t_6)) - t_5;
}

Julia

function code(cosTheta, alpha)
	t_0 = fma(Float32(-1.0), Float32(cosTheta * cosTheta), Float32(cosTheta * cosTheta))
	t_1 = Float32(Float32(1.0) - Float32(cosTheta * cosTheta))
	t_2 = Float32(Float32(pi) * Float32(log(alpha) * (t_1 ^ Float32(2.0))))
	t_3 = Float32(t_0 / t_2)
	t_4 = Float32(Float32(1.0) / Float32(Float32(pi) * Float32(log(alpha) * t_1)))
	t_5 = Float32(Float32(0.5) * t_4)
	t_6 = Float32(Float32(-0.5) * Float32(Float32(cosTheta * cosTheta) / t_2))
	t_7 = Float32(t_5 - t_6)
	t_8 = fma(Float32(0.5), t_3, Float32(Float32(Float32(cosTheta * cosTheta) * t_7) / t_1))
	return Float32(Float32(Float32(alpha * alpha) * Float32(fma(Float32(0.5), t_4, Float32(Float32(alpha * alpha) * Float32(Float32(Float32(Float32(-1.0) * Float32(alpha * alpha)) * fma(Float32(-1.0), Float32(Float32(t_0 * t_7) / t_1), fma(Float32(-1.0), Float32(Float32(Float32(cosTheta * cosTheta) * t_8) / t_1), Float32(Float32(-0.5) * t_3)))) - t_8))) - t_6)) - t_5)
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. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. flip3-- N/A

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

   5. lower-/.f32 N/A

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

   6. pow-pow N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\alpha}^{\color{blue}{6}} - {1}^{3}}{{\alpha}^{2} \cdot {\alpha}^{2} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\alpha}^{6} - \color{blue}{1}}{{\alpha}^{2} \cdot {\alpha}^{2} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{\color{blue}{{\alpha}^{6} - 1}}{{\alpha}^{2} \cdot {\alpha}^{2} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. lower-pow.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 N/A

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

   15. sqr-neg-rev N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\color{blue}{\left(\mathsf{neg}\left({\alpha}^{2}\right)\right) \cdot \left(\mathsf{neg}\left({\alpha}^{2}\right)\right)} + \left(1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   16. lower-fma.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left({\alpha}^{2}\right), \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)}} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   17. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(\color{blue}{-{\alpha}^{2}}, \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   18. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\color{blue}{\alpha \cdot \alpha}, \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   19. lift-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\color{blue}{\alpha \cdot \alpha}, \mathsf{neg}\left({\alpha}^{2}\right), 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   20. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, \color{blue}{-{\alpha}^{2}}, 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   21. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\color{blue}{\alpha \cdot \alpha}, 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   22. lift-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\color{blue}{\alpha \cdot \alpha}, 1 \cdot 1 + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

   23. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\alpha \cdot \alpha, \color{blue}{1} + {\alpha}^{2} \cdot 1\right)} \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\color{blue}{\frac{{\left(\alpha \cdot \alpha\right)}^{3} - 1}{\mathsf{fma}\left(-\alpha \cdot \alpha, -\alpha \cdot \alpha, 1 + \left(\alpha \cdot \alpha\right) \cdot 1\right)}} \cdot cosTheta\right) \cdot cosTheta\right)} \]

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

   \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\pi \cdot \left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)}, \left(\alpha \cdot \alpha\right) \cdot \left(\left(-1 \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, cosTheta \cdot cosTheta, cosTheta \cdot cosTheta\right) \cdot \left(0.5 \cdot \frac{1}{\pi \cdot \left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} - -0.5 \cdot \frac{cosTheta \cdot cosTheta}{\pi \cdot \left(\log \alpha \cdot {\left(1 - cosTheta \cdot cosTheta\right)}^{2}\right)}\right)}{1 - cosTheta \cdot cosTheta}, \mathsf{fma}\left(-1, \frac{\left(cosTheta \cdot cosTheta\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, cosTheta \cdot cosTheta, cosTheta \cdot cosTheta\right)}{\pi \cdot \left(\log \alpha \cdot {\left(1 - cosTheta \cdot cosTheta\right)}^{2}\right)}, \frac{\left(cosTheta \cdot cosTheta\right) \cdot \left(0.5 \cdot \frac{1}{\pi \cdot \left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} - -0.5 \cdot \frac{cosTheta \cdot cosTheta}{\pi \cdot \left(\log \alpha \cdot {\left(1 - cosTheta \cdot cosTheta\right)}^{2}\right)}\right)}{1 - cosTheta \cdot cosTheta}\right)}{1 - cosTheta \cdot cosTheta}, -0.5 \cdot \frac{\mathsf{fma}\left(-1, cosTheta \cdot cosTheta, cosTheta \cdot cosTheta\right)}{\pi \cdot \left(\log \alpha \cdot {\left(1 - cosTheta \cdot cosTheta\right)}^{2}\right)}\right)\right) - \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, cosTheta \cdot cosTheta, cosTheta \cdot cosTheta\right)}{\pi \cdot \left(\log \alpha \cdot {\left(1 - cosTheta \cdot cosTheta\right)}^{2}\right)}, \frac{\left(cosTheta \cdot cosTheta\right) \cdot \left(0.5 \cdot \frac{1}{\pi \cdot \left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} - -0.5 \cdot \frac{cosTheta \cdot cosTheta}{\pi \cdot \left(\log \alpha \cdot {\left(1 - cosTheta \cdot cosTheta\right)}^{2}\right)}\right)}{1 - cosTheta \cdot cosTheta}\right)\right)\right) - -0.5 \cdot \frac{cosTheta \cdot cosTheta}{\pi \cdot \left(\log \alpha \cdot {\left(1 - cosTheta \cdot cosTheta\right)}^{2}\right)}\right) - 0.5 \cdot \frac{1}{\pi \cdot \left(\log \alpha \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} \]
9. Add Preprocessing

Alternative 6: 98.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \alpha \cdot \pi\\ t_1 := \mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\\ t_2 := \frac{1}{t\_1}\\ t_3 := {\left(\alpha \cdot cosTheta\right)}^{1}\\ t_4 := \frac{0.5}{t\_0} \cdot t\_2 - \frac{-0.5 \cdot \left(cosTheta \cdot cosTheta\right)}{t\_0 \cdot {t\_1}^{2}}\\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(t\_3 \cdot t\_3\right) \cdot t\_4}{t\_1}, -1, t\_4\right), \alpha \cdot \alpha, -0.5 \cdot \left(\frac{1}{t\_0} \cdot t\_2\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (* (log alpha) PI))
        (t_1 (fma (* cosTheta cosTheta) -1.0 1.0))
        (t_2 (/ 1.0 t_1))
        (t_3 (pow (* alpha cosTheta) 1.0))
        (t_4
         (-
          (* (/ 0.5 t_0) t_2)
          (/ (* -0.5 (* cosTheta cosTheta)) (* t_0 (pow t_1 2.0))))))
   (fma
    (fma (/ (* (* t_3 t_3) t_4) t_1) -1.0 t_4)
    (* alpha alpha)
    (* -0.5 (* (/ 1.0 t_0) t_2)))))

C

float code(float cosTheta, float alpha) {
	float t_0 = logf(alpha) * ((float) M_PI);
	float t_1 = fmaf((cosTheta * cosTheta), -1.0f, 1.0f);
	float t_2 = 1.0f / t_1;
	float t_3 = powf((alpha * cosTheta), 1.0f);
	float t_4 = ((0.5f / t_0) * t_2) - ((-0.5f * (cosTheta * cosTheta)) / (t_0 * powf(t_1, 2.0f)));
	return fmaf(fmaf((((t_3 * t_3) * t_4) / t_1), -1.0f, t_4), (alpha * alpha), (-0.5f * ((1.0f / t_0) * t_2)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(log(alpha) * Float32(pi))
	t_1 = fma(Float32(cosTheta * cosTheta), Float32(-1.0), Float32(1.0))
	t_2 = Float32(Float32(1.0) / t_1)
	t_3 = Float32(alpha * cosTheta) ^ Float32(1.0)
	t_4 = Float32(Float32(Float32(Float32(0.5) / t_0) * t_2) - Float32(Float32(Float32(-0.5) * Float32(cosTheta * cosTheta)) / Float32(t_0 * (t_1 ^ Float32(2.0)))))
	return fma(fma(Float32(Float32(Float32(t_3 * t_3) * t_4) / t_1), Float32(-1.0), t_4), Float32(alpha * alpha), Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) / t_0) * t_2)))
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. Add Preprocessing

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left({\left(\alpha \cdot cosTheta\right)}^{1} \cdot {\left(\alpha \cdot cosTheta\right)}^{1}\right) \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5 \cdot \left(cosTheta \cdot cosTheta\right)}{\left(\log \alpha \cdot \pi\right) \cdot {\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5 \cdot \left(cosTheta \cdot cosTheta\right)}{\left(\log \alpha \cdot \pi\right) \cdot {\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right), \alpha \cdot \alpha, -0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right)} \]
6. Add Preprocessing

Alternative 7: 97.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\\ t_1 := \log \alpha \cdot \pi\\ t_2 := \frac{1}{t\_0}\\ t_3 := \frac{0.5}{t\_1} \cdot t\_2 - \frac{-0.5}{t\_1} \cdot \frac{cosTheta \cdot cosTheta}{{t\_0}^{2}}\\ t_4 := -0.5 \cdot \left(\frac{1}{t\_1} \cdot t\_2\right)\\ t_5 := \mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot t\_3}{t\_0}, -1, t\_3\right) \cdot \left(\alpha \cdot \alpha\right)\\ \frac{{t\_5}^{3} + {t\_4}^{3}}{t\_5 \cdot t\_5 + \left(t\_4 \cdot t\_4 - t\_5 \cdot t\_4\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (fma (* cosTheta cosTheta) -1.0 1.0))
        (t_1 (* (log alpha) PI))
        (t_2 (/ 1.0 t_0))
        (t_3
         (-
          (* (/ 0.5 t_1) t_2)
          (* (/ -0.5 t_1) (/ (* cosTheta cosTheta) (pow t_0 2.0)))))
        (t_4 (* -0.5 (* (/ 1.0 t_1) t_2)))
        (t_5
         (*
          (fma (/ (* (pow (* alpha cosTheta) 2.0) t_3) t_0) -1.0 t_3)
          (* alpha alpha))))
   (/
    (+ (pow t_5 3.0) (pow t_4 3.0))
    (+ (* t_5 t_5) (- (* t_4 t_4) (* t_5 t_4))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = fmaf((cosTheta * cosTheta), -1.0f, 1.0f);
	float t_1 = logf(alpha) * ((float) M_PI);
	float t_2 = 1.0f / t_0;
	float t_3 = ((0.5f / t_1) * t_2) - ((-0.5f / t_1) * ((cosTheta * cosTheta) / powf(t_0, 2.0f)));
	float t_4 = -0.5f * ((1.0f / t_1) * t_2);
	float t_5 = fmaf(((powf((alpha * cosTheta), 2.0f) * t_3) / t_0), -1.0f, t_3) * (alpha * alpha);
	return (powf(t_5, 3.0f) + powf(t_4, 3.0f)) / ((t_5 * t_5) + ((t_4 * t_4) - (t_5 * t_4)));
}

Julia

function code(cosTheta, alpha)
	t_0 = fma(Float32(cosTheta * cosTheta), Float32(-1.0), Float32(1.0))
	t_1 = Float32(log(alpha) * Float32(pi))
	t_2 = Float32(Float32(1.0) / t_0)
	t_3 = Float32(Float32(Float32(Float32(0.5) / t_1) * t_2) - Float32(Float32(Float32(-0.5) / t_1) * Float32(Float32(cosTheta * cosTheta) / (t_0 ^ Float32(2.0)))))
	t_4 = Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) / t_1) * t_2))
	t_5 = Float32(fma(Float32(Float32((Float32(alpha * cosTheta) ^ Float32(2.0)) * t_3) / t_0), Float32(-1.0), t_3) * Float32(alpha * alpha))
	return Float32(Float32((t_5 ^ Float32(3.0)) + (t_4 ^ Float32(3.0))) / Float32(Float32(t_5 * t_5) + Float32(Float32(t_4 * t_4) - Float32(t_5 * t_4))))
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. Add Preprocessing

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 97.8%

   \[\leadsto \frac{{\left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right)}^{3} + {\left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right) + \left(\left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right) - \left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right)\right)}} \]
8. Add Preprocessing

Alternative 8: 97.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\log \alpha}^{3}\\ t_1 := \frac{\alpha \cdot \alpha}{\pi}\\ t_2 := \pi \cdot \log \alpha\\ t_3 := \frac{\alpha \cdot \alpha}{t\_2}\\ t_4 := \left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}\\ t_5 := \frac{1}{t\_4}\\ t_6 := --1 \cdot -1\\ t_7 := --1 \cdot -0.5\\ t_8 := t\_3 - t\_7 \cdot t\_3\\ t_9 := {\pi}^{3} \cdot t\_0\\ t_10 := 0.125 \cdot \frac{{\alpha}^{6}}{t\_9} - 0.125 \cdot \frac{1}{t\_9}\\ t_11 := \frac{1}{t\_2}\\ t_12 := \mathsf{fma}\left(-1, t\_8, 1.5 \cdot t\_11\right)\\ t_13 := \mathsf{fma}\left(-0.5, t\_3, t\_11\right)\\ t_14 := \frac{t\_12}{\log \alpha}\\ t_15 := t\_13 \cdot t\_13\\ t_16 := \mathsf{fma}\left(-1, \frac{t\_8}{t\_2}, \mathsf{fma}\left(1.5, t\_5, t\_15\right)\right)\\ t_17 := \frac{\alpha \cdot \alpha}{t\_4}\\ t_18 := \mathsf{fma}\left(0.25, t\_5, 0.25 \cdot \frac{{\alpha}^{4}}{t\_4}\right) - -0.25 \cdot t\_17\\ t_19 := \frac{\left(\alpha \cdot \alpha\right) \cdot t\_13}{t\_2} - t\_7 \cdot t\_17\\ t_20 := \mathsf{fma}\left(0.75, t\_5, {\alpha}^{4} \cdot t\_16\right) - -0.5 \cdot \left(t\_1 \cdot t\_14 - t\_6 \cdot t\_19\right)\\ t_21 := \frac{{\alpha}^{6} \cdot \left(0.25 \cdot \frac{t\_12}{t\_4} + \mathsf{fma}\left(0.5, \frac{t\_16}{t\_2}, \frac{t\_15}{t\_2}\right)\right)}{t\_18}\\ t_22 := {\pi}^{3} \cdot \left(t\_0 \cdot t\_18\right)\\ t_23 := \frac{1}{t\_22}\\ t_24 := t\_18 \cdot t\_18\\ t_25 := 1.5 \cdot t\_3 - t\_6 \cdot t\_8\\ t_26 := \mathsf{fma}\left(-1, \frac{t\_25}{t\_2}, \mathsf{fma}\left(2, t\_12 \cdot t\_13, 2 \cdot t\_5\right)\right)\\ t_27 := \mathsf{fma}\left(-1, t\_25, 2 \cdot t\_11\right)\\ t_28 := \mathsf{fma}\left(0.5, t\_5, \frac{{\alpha}^{4} \cdot t\_13}{t\_2}\right) - -0.5 \cdot t\_19\\ t_29 := \mathsf{fma}\left(0.375, t\_23, \frac{t\_10 \cdot t\_28}{t\_24}\right)\\ t_30 := \frac{t\_13}{t\_4}\\ t_31 := \frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, t\_30, 0.5 \cdot t\_30\right)}{t\_18}\\ t_32 := t\_31 - t\_29\\ t_33 := \mathsf{fma}\left(0.75, t\_23, \frac{\mathsf{fma}\left(t\_10 \cdot t\_20, t\_18, t\_24 \cdot \left(t\_28 \cdot t\_32\right)\right)}{t\_24 \cdot t\_18}\right)\\ \mathsf{fma}\left(0.125, \frac{{\alpha}^{6}}{t\_22}, \left(cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{t\_27}{t\_4}, \mathsf{fma}\left(0.5, \frac{t\_26}{t\_2}, \mathsf{fma}\left(t\_16, t\_13, \frac{t\_12}{\pi} \cdot \frac{t\_13}{\log \alpha}\right)\right)\right)}{t\_18} - \mathsf{fma}\left(1.25, t\_23, \frac{t\_10 \cdot \left(\mathsf{fma}\left({\alpha}^{4}, t\_26, t\_5\right) - -0.5 \cdot \left(t\_1 \cdot \frac{t\_27}{\log \alpha} - \left(\left(-t\_1\right) \cdot t\_14 - t\_19\right)\right)\right)}{t\_24} + \frac{\mathsf{fma}\left(t\_28, t\_21 - t\_33, t\_20 \cdot t\_32\right)}{t\_18}\right), t\_21\right) - t\_33, t\_31\right) - t\_29\right)\right) - \frac{0.125}{t\_22} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (pow (log alpha) 3.0))
        (t_1 (/ (* alpha alpha) PI))
        (t_2 (* PI (log alpha)))
        (t_3 (/ (* alpha alpha) t_2))
        (t_4 (* (* PI PI) (pow (log alpha) 2.0)))
        (t_5 (/ 1.0 t_4))
        (t_6 (- (* -1.0 -1.0)))
        (t_7 (- (* -1.0 -0.5)))
        (t_8 (- t_3 (* t_7 t_3)))
        (t_9 (* (pow PI 3.0) t_0))
        (t_10 (- (* 0.125 (/ (pow alpha 6.0) t_9)) (* 0.125 (/ 1.0 t_9))))
        (t_11 (/ 1.0 t_2))
        (t_12 (fma -1.0 t_8 (* 1.5 t_11)))
        (t_13 (fma -0.5 t_3 t_11))
        (t_14 (/ t_12 (log alpha)))
        (t_15 (* t_13 t_13))
        (t_16 (fma -1.0 (/ t_8 t_2) (fma 1.5 t_5 t_15)))
        (t_17 (/ (* alpha alpha) t_4))
        (t_18
         (- (fma 0.25 t_5 (* 0.25 (/ (pow alpha 4.0) t_4))) (* -0.25 t_17)))
        (t_19 (- (/ (* (* alpha alpha) t_13) t_2) (* t_7 t_17)))
        (t_20
         (-
          (fma 0.75 t_5 (* (pow alpha 4.0) t_16))
          (* -0.5 (- (* t_1 t_14) (* t_6 t_19)))))
        (t_21
         (/
          (*
           (pow alpha 6.0)
           (+ (* 0.25 (/ t_12 t_4)) (fma 0.5 (/ t_16 t_2) (/ t_15 t_2))))
          t_18))
        (t_22 (* (pow PI 3.0) (* t_0 t_18)))
        (t_23 (/ 1.0 t_22))
        (t_24 (* t_18 t_18))
        (t_25 (- (* 1.5 t_3) (* t_6 t_8)))
        (t_26 (fma -1.0 (/ t_25 t_2) (fma 2.0 (* t_12 t_13) (* 2.0 t_5))))
        (t_27 (fma -1.0 t_25 (* 2.0 t_11)))
        (t_28 (- (fma 0.5 t_5 (/ (* (pow alpha 4.0) t_13) t_2)) (* -0.5 t_19)))
        (t_29 (fma 0.375 t_23 (/ (* t_10 t_28) t_24)))
        (t_30 (/ t_13 t_4))
        (t_31 (/ (* (pow alpha 6.0) (fma 0.25 t_30 (* 0.5 t_30))) t_18))
        (t_32 (- t_31 t_29))
        (t_33
         (fma
          0.75
          t_23
          (/ (fma (* t_10 t_20) t_18 (* t_24 (* t_28 t_32))) (* t_24 t_18)))))
   (-
    (fma
     0.125
     (/ (pow alpha 6.0) t_22)
     (*
      (* cosTheta cosTheta)
      (-
       (fma
        (* cosTheta cosTheta)
        (-
         (fma
          (* cosTheta cosTheta)
          (-
           (/
            (*
             (pow alpha 6.0)
             (fma
              0.25
              (/ t_27 t_4)
              (fma
               0.5
               (/ t_26 t_2)
               (fma t_16 t_13 (* (/ t_12 PI) (/ t_13 (log alpha)))))))
            t_18)
           (fma
            1.25
            t_23
            (+
             (/
              (*
               t_10
               (-
                (fma (pow alpha 4.0) t_26 t_5)
                (*
                 -0.5
                 (- (* t_1 (/ t_27 (log alpha))) (- (* (- t_1) t_14) t_19)))))
              t_24)
             (/ (fma t_28 (- t_21 t_33) (* t_20 t_32)) t_18))))
          t_21)
         t_33)
        t_31)
       t_29)))
    (/ 0.125 t_22))))

C

float code(float cosTheta, float alpha) {
	float t_0 = powf(logf(alpha), 3.0f);
	float t_1 = (alpha * alpha) / ((float) M_PI);
	float t_2 = ((float) M_PI) * logf(alpha);
	float t_3 = (alpha * alpha) / t_2;
	float t_4 = (((float) M_PI) * ((float) M_PI)) * powf(logf(alpha), 2.0f);
	float t_5 = 1.0f / t_4;
	float t_6 = -(-1.0f * -1.0f);
	float t_7 = -(-1.0f * -0.5f);
	float t_8 = t_3 - (t_7 * t_3);
	float t_9 = powf(((float) M_PI), 3.0f) * t_0;
	float t_10 = (0.125f * (powf(alpha, 6.0f) / t_9)) - (0.125f * (1.0f / t_9));
	float t_11 = 1.0f / t_2;
	float t_12 = fmaf(-1.0f, t_8, (1.5f * t_11));
	float t_13 = fmaf(-0.5f, t_3, t_11);
	float t_14 = t_12 / logf(alpha);
	float t_15 = t_13 * t_13;
	float t_16 = fmaf(-1.0f, (t_8 / t_2), fmaf(1.5f, t_5, t_15));
	float t_17 = (alpha * alpha) / t_4;
	float t_18 = fmaf(0.25f, t_5, (0.25f * (powf(alpha, 4.0f) / t_4))) - (-0.25f * t_17);
	float t_19 = (((alpha * alpha) * t_13) / t_2) - (t_7 * t_17);
	float t_20 = fmaf(0.75f, t_5, (powf(alpha, 4.0f) * t_16)) - (-0.5f * ((t_1 * t_14) - (t_6 * t_19)));
	float t_21 = (powf(alpha, 6.0f) * ((0.25f * (t_12 / t_4)) + fmaf(0.5f, (t_16 / t_2), (t_15 / t_2)))) / t_18;
	float t_22 = powf(((float) M_PI), 3.0f) * (t_0 * t_18);
	float t_23 = 1.0f / t_22;
	float t_24 = t_18 * t_18;
	float t_25 = (1.5f * t_3) - (t_6 * t_8);
	float t_26 = fmaf(-1.0f, (t_25 / t_2), fmaf(2.0f, (t_12 * t_13), (2.0f * t_5)));
	float t_27 = fmaf(-1.0f, t_25, (2.0f * t_11));
	float t_28 = fmaf(0.5f, t_5, ((powf(alpha, 4.0f) * t_13) / t_2)) - (-0.5f * t_19);
	float t_29 = fmaf(0.375f, t_23, ((t_10 * t_28) / t_24));
	float t_30 = t_13 / t_4;
	float t_31 = (powf(alpha, 6.0f) * fmaf(0.25f, t_30, (0.5f * t_30))) / t_18;
	float t_32 = t_31 - t_29;
	float t_33 = fmaf(0.75f, t_23, (fmaf((t_10 * t_20), t_18, (t_24 * (t_28 * t_32))) / (t_24 * t_18)));
	return fmaf(0.125f, (powf(alpha, 6.0f) / t_22), ((cosTheta * cosTheta) * (fmaf((cosTheta * cosTheta), (fmaf((cosTheta * cosTheta), (((powf(alpha, 6.0f) * fmaf(0.25f, (t_27 / t_4), fmaf(0.5f, (t_26 / t_2), fmaf(t_16, t_13, ((t_12 / ((float) M_PI)) * (t_13 / logf(alpha))))))) / t_18) - fmaf(1.25f, t_23, (((t_10 * (fmaf(powf(alpha, 4.0f), t_26, t_5) - (-0.5f * ((t_1 * (t_27 / logf(alpha))) - ((-t_1 * t_14) - t_19))))) / t_24) + (fmaf(t_28, (t_21 - t_33), (t_20 * t_32)) / t_18)))), t_21) - t_33), t_31) - t_29))) - (0.125f / t_22);
}

Julia

function code(cosTheta, alpha)
	t_0 = log(alpha) ^ Float32(3.0)
	t_1 = Float32(Float32(alpha * alpha) / Float32(pi))
	t_2 = Float32(Float32(pi) * log(alpha))
	t_3 = Float32(Float32(alpha * alpha) / t_2)
	t_4 = Float32(Float32(Float32(pi) * Float32(pi)) * (log(alpha) ^ Float32(2.0)))
	t_5 = Float32(Float32(1.0) / t_4)
	t_6 = Float32(-Float32(Float32(-1.0) * Float32(-1.0)))
	t_7 = Float32(-Float32(Float32(-1.0) * Float32(-0.5)))
	t_8 = Float32(t_3 - Float32(t_7 * t_3))
	t_9 = Float32((Float32(pi) ^ Float32(3.0)) * t_0)
	t_10 = Float32(Float32(Float32(0.125) * Float32((alpha ^ Float32(6.0)) / t_9)) - Float32(Float32(0.125) * Float32(Float32(1.0) / t_9)))
	t_11 = Float32(Float32(1.0) / t_2)
	t_12 = fma(Float32(-1.0), t_8, Float32(Float32(1.5) * t_11))
	t_13 = fma(Float32(-0.5), t_3, t_11)
	t_14 = Float32(t_12 / log(alpha))
	t_15 = Float32(t_13 * t_13)
	t_16 = fma(Float32(-1.0), Float32(t_8 / t_2), fma(Float32(1.5), t_5, t_15))
	t_17 = Float32(Float32(alpha * alpha) / t_4)
	t_18 = Float32(fma(Float32(0.25), t_5, Float32(Float32(0.25) * Float32((alpha ^ Float32(4.0)) / t_4))) - Float32(Float32(-0.25) * t_17))
	t_19 = Float32(Float32(Float32(Float32(alpha * alpha) * t_13) / t_2) - Float32(t_7 * t_17))
	t_20 = Float32(fma(Float32(0.75), t_5, Float32((alpha ^ Float32(4.0)) * t_16)) - Float32(Float32(-0.5) * Float32(Float32(t_1 * t_14) - Float32(t_6 * t_19))))
	t_21 = Float32(Float32((alpha ^ Float32(6.0)) * Float32(Float32(Float32(0.25) * Float32(t_12 / t_4)) + fma(Float32(0.5), Float32(t_16 / t_2), Float32(t_15 / t_2)))) / t_18)
	t_22 = Float32((Float32(pi) ^ Float32(3.0)) * Float32(t_0 * t_18))
	t_23 = Float32(Float32(1.0) / t_22)
	t_24 = Float32(t_18 * t_18)
	t_25 = Float32(Float32(Float32(1.5) * t_3) - Float32(t_6 * t_8))
	t_26 = fma(Float32(-1.0), Float32(t_25 / t_2), fma(Float32(2.0), Float32(t_12 * t_13), Float32(Float32(2.0) * t_5)))
	t_27 = fma(Float32(-1.0), t_25, Float32(Float32(2.0) * t_11))
	t_28 = Float32(fma(Float32(0.5), t_5, Float32(Float32((alpha ^ Float32(4.0)) * t_13) / t_2)) - Float32(Float32(-0.5) * t_19))
	t_29 = fma(Float32(0.375), t_23, Float32(Float32(t_10 * t_28) / t_24))
	t_30 = Float32(t_13 / t_4)
	t_31 = Float32(Float32((alpha ^ Float32(6.0)) * fma(Float32(0.25), t_30, Float32(Float32(0.5) * t_30))) / t_18)
	t_32 = Float32(t_31 - t_29)
	t_33 = fma(Float32(0.75), t_23, Float32(fma(Float32(t_10 * t_20), t_18, Float32(t_24 * Float32(t_28 * t_32))) / Float32(t_24 * t_18)))
	return Float32(fma(Float32(0.125), Float32((alpha ^ Float32(6.0)) / t_22), Float32(Float32(cosTheta * cosTheta) * Float32(fma(Float32(cosTheta * cosTheta), Float32(fma(Float32(cosTheta * cosTheta), Float32(Float32(Float32((alpha ^ Float32(6.0)) * fma(Float32(0.25), Float32(t_27 / t_4), fma(Float32(0.5), Float32(t_26 / t_2), fma(t_16, t_13, Float32(Float32(t_12 / Float32(pi)) * Float32(t_13 / log(alpha))))))) / t_18) - fma(Float32(1.25), t_23, Float32(Float32(Float32(t_10 * Float32(fma((alpha ^ Float32(4.0)), t_26, t_5) - Float32(Float32(-0.5) * Float32(Float32(t_1 * Float32(t_27 / log(alpha))) - Float32(Float32(Float32(-t_1) * t_14) - t_19))))) / t_24) + Float32(fma(t_28, Float32(t_21 - t_33), Float32(t_20 * t_32)) / t_18)))), t_21) - t_33), t_31) - t_29))) - Float32(Float32(0.125) / t_22))
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. Add Preprocessing

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 98.2%

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

7. Applied rewrites 97.8%

   \[\leadsto \frac{{\left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right)}^{3} + {\left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right) + \left(\left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right) - \left(\mathsf{fma}\left(\frac{{\left(\alpha \cdot cosTheta\right)}^{2} \cdot \left(\frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}, -1, \frac{0.5}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)} - \frac{-0.5}{\log \alpha \cdot \pi} \cdot \frac{cosTheta \cdot cosTheta}{{\left(\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)\right)}^{2}}\right) \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(-0.5 \cdot \left(\frac{1}{\log \alpha \cdot \pi} \cdot \frac{1}{\mathsf{fma}\left(cosTheta \cdot cosTheta, -1, 1\right)}\right)\right)\right)}} \]

8. Taylor expanded in cosTheta around 0

   \[\leadsto \left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + {cosTheta}^{2} \cdot \left(\left({cosTheta}^{2} \cdot \left(\left({cosTheta}^{2} \cdot \left(\frac{{\alpha}^{6} \cdot \left(\frac{1}{4} \cdot \frac{-1 \cdot \left(\frac{3}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right) + 2 \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{3}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(2 \cdot \left(\left(-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right) + 2 \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(\left(-1 \cdot \frac{\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(\frac{3}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{\left(-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right)\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}} - \left(\frac{5}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + \left(\frac{\left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}} - \frac{1}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\left({\alpha}^{4} \cdot \left(-1 \cdot \frac{\frac{3}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(2 \cdot \left(\left(-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right) + 2 \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right) + \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(-1 \cdot \left(\frac{3}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right) + 2 \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2} \cdot \left(-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)\right)\right)}{{\left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}^{2}} + \left(\frac{\left(\left(\frac{1}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{{\alpha}^{4} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \left(\frac{1}{4} \cdot \frac{-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(\frac{3}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{{\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}} - \left(\frac{3}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + \left(\frac{\left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}} - \frac{1}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\left(\frac{3}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\alpha}^{4} \cdot \left(-1 \cdot \frac{\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(\frac{3}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}\right)\right)\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)\right)}{{\left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}^{2}} + \frac{\left(\left(\frac{1}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{{\alpha}^{4} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \left(\frac{1}{4} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{2} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}} - \left(\frac{3}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + \frac{\left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}} - \frac{1}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{{\alpha}^{4} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)}{{\left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}^{2}}\right)\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}}\right)\right)\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}} + \frac{\left(\left(\frac{3}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\alpha}^{4} \cdot \left(-1 \cdot \frac{\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(\frac{3}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}\right)\right)\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \left(\frac{1}{4} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{2} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}} - \left(\frac{3}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + \frac{\left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}} - \frac{1}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{{\alpha}^{4} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)}{{\left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}^{2}}\right)\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}}\right)\right)\right)\right) + \frac{{\alpha}^{6} \cdot \left(\frac{1}{4} \cdot \frac{-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \left(\frac{1}{2} \cdot \frac{-1 \cdot \frac{\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(\frac{3}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{{\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}}\right) - \left(\frac{3}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + \left(\frac{\left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}} - \frac{1}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\left(\frac{3}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\alpha}^{4} \cdot \left(-1 \cdot \frac{\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \left(\frac{3}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + {\left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}^{2}\right)\right)\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(-1 \cdot \left(\frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) + \frac{3}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - -1 \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)\right)}{{\left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}^{2}} + \frac{\left(\left(\frac{1}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{{\alpha}^{4} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \left(\frac{1}{4} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{2} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}} - \left(\frac{3}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + \frac{\left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}} - \frac{1}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{{\alpha}^{4} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)}{{\left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}^{2}}\right)\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}}\right)\right)\right) + \frac{{\alpha}^{6} \cdot \left(\frac{1}{4} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{2} \cdot \frac{\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}{\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}}\right) - \left(\frac{3}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)} + \frac{\left(\frac{1}{8} \cdot \frac{{\alpha}^{6}}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}} - \frac{1}{8} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{{\alpha}^{4} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right) - \frac{-1}{2} \cdot \left(\frac{{\alpha}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\alpha}^{2}}{\mathsf{PI}\left(\right) \cdot \log \alpha} + \frac{1}{\mathsf{PI}\left(\right) \cdot \log \alpha}\right)}{\mathsf{PI}\left(\right) \cdot \log \alpha} - \frac{-1}{2} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)}{{\left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)}^{2}}\right)\right)\right) - \color{blue}{\frac{\frac{1}{8}}{{\mathsf{PI}\left(\right)}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\left(\frac{1}{4} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}} + \frac{1}{4} \cdot \frac{{\alpha}^{4}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right) - \frac{-1}{4} \cdot \frac{{\alpha}^{2}}{{\mathsf{PI}\left(\right)}^{2} \cdot {\log \alpha}^{2}}\right)\right)}} \]

10. Applied rewrites 97.2%

    \[\leadsto \mathsf{fma}\left(0.125, \frac{{\alpha}^{6}}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \left(cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-1, 1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -1 \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, \frac{1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -1 \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}, \mathsf{fma}\left(2, \mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\pi \cdot \log \alpha}, \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right), \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right), \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\pi} \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha}\right)\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(1.25, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left({\alpha}^{4}, \mathsf{fma}\left(-1, \frac{1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -1 \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}, \mathsf{fma}\left(2, \mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right), \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, 1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -1 \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - -1 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - -1 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right), \frac{{\alpha}^{6} \cdot \left(0.25 \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}} + \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)}{\pi \cdot \log \alpha}, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.75, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\mathsf{fma}\left(\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.75, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, {\alpha}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - -1 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right), \mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right)\right)}{\left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right), \left(\mathsf{fma}\left(0.75, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, {\alpha}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - -1 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}}\right), \frac{{\alpha}^{6} \cdot \left(0.25 \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}} + \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)}{\pi \cdot \log \alpha}, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}}\right) - \mathsf{fma}\left(0.75, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\mathsf{fma}\left(\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.75, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, {\alpha}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - -1 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right), \mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right)\right)}{\left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right), \frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}}\right) - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - -0.5 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right) - \color{blue}{\frac{0.125}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}} \]

11. Final simplification 97.2%

    \[\leadsto \mathsf{fma}\left(0.125, \frac{{\alpha}^{6}}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \left(cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{fma}\left(cosTheta \cdot cosTheta, \mathsf{fma}\left(cosTheta \cdot cosTheta, \frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-1, 1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -1\right) \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, \frac{1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -1\right) \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}, \mathsf{fma}\left(2, \mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\pi \cdot \log \alpha}, \mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right), \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right), \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\pi} \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha}\right)\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(1.25, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left({\alpha}^{4}, \mathsf{fma}\left(-1, \frac{1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -1\right) \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}, \mathsf{fma}\left(2, \mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right), \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, 1.5 \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -1\right) \cdot \left(\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}\right), 2 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - \left(\left(-\frac{\alpha \cdot \alpha}{\pi}\right) \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right), \frac{{\alpha}^{6} \cdot \left(0.25 \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}} + \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)}{\pi \cdot \log \alpha}, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.75, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\mathsf{fma}\left(\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.75, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, {\alpha}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - \left(--1 \cdot -1\right) \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right), \mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right)\right)}{\left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right), \left(\mathsf{fma}\left(0.75, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, {\alpha}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - \left(--1 \cdot -1\right) \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}}\right), \frac{{\alpha}^{6} \cdot \left(0.25 \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}} + \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)}{\pi \cdot \log \alpha}, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right)\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}}\right) - \mathsf{fma}\left(0.75, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\mathsf{fma}\left(\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.75, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, {\alpha}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}}{\pi \cdot \log \alpha}, \mathsf{fma}\left(1.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)\right)\right)\right) - -0.5 \cdot \left(\frac{\alpha \cdot \alpha}{\pi} \cdot \frac{\mathsf{fma}\left(-1, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, 1.5 \cdot \frac{1}{\pi \cdot \log \alpha}\right)}{\log \alpha} - \left(--1 \cdot -1\right) \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)\right), \mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}} - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right)\right)}{\left(\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right), \frac{{\alpha}^{6} \cdot \mathsf{fma}\left(0.25, \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}{\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}}\right) - \mathsf{fma}\left(0.375, \frac{1}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}, \frac{\left(0.125 \cdot \frac{{\alpha}^{6}}{{\pi}^{3} \cdot {\log \alpha}^{3}} - 0.125 \cdot \frac{1}{{\pi}^{3} \cdot {\log \alpha}^{3}}\right) \cdot \left(\mathsf{fma}\left(0.5, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, \frac{{\alpha}^{4} \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha}\right) - -0.5 \cdot \left(\frac{\left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(-0.5, \frac{\alpha \cdot \alpha}{\pi \cdot \log \alpha}, \frac{1}{\pi \cdot \log \alpha}\right)}{\pi \cdot \log \alpha} - \left(--1 \cdot -0.5\right) \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)}{\left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)}\right)\right)\right) - \frac{0.125}{{\pi}^{3} \cdot \left({\log \alpha}^{3} \cdot \left(\mathsf{fma}\left(0.25, \frac{1}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}, 0.25 \cdot \frac{{\alpha}^{4}}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right) - -0.25 \cdot \frac{\alpha \cdot \alpha}{\left(\pi \cdot \pi\right) \cdot {\log \alpha}^{2}}\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(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)))))