GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 23.8s

Alternatives: 1

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	t_1 = Float32(alpha * alpha) ^ Float32(pi)
	return Float32(fma(Float32(fma(t_0, Float32(alpha * alpha), Float32(1.0)) - fma(log(Float32((t_1 ^ Float32(alpha * alpha)) / t_1)), Float32(t_0 / log(((alpha ^ Float32(pi)) ^ Float32(2.0)))), Float32(alpha * alpha))), Float32(cosTheta * cosTheta), t_0) / Float32(Float32(Float32(1.0) + Float32((Float32(alpha * cosTheta) ^ Float32(2.0)) - Float32(cosTheta * cosTheta))) * log(t_1)))
end

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.4%

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

   1. lift-pow.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. pow2 N/A

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

   10. +-commutative N/A

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

   11. pow2 N/A

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

   12. pow2 N/A

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

   13. fp-cancel-sign-sub-inv N/A

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

   14. pow-sub N/A

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

6. Applied rewrites 98.4%

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

7. Taylor expanded in cosTheta around inf

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in cosTheta around 0

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

12. Applied rewrites 98.7%

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

Reproduce

herbie shell --seed 2025059 
(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)))))