GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 8.9s

Alternatives: 5

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

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	t_1 = log(((alpha ^ Float32(pi)) ^ Float32(2.0)))
	return Float32(t_0 / fma(t_1, Float32(1.0), Float32(t_1 * Float32(Float32(cosTheta * cosTheta) * t_0))))
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 98.7%

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

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

Math

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

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 98.5%

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

Alternative 3: 98.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	t_0 = log(Float32(alpha * alpha))
	t_1 = fma(Float32(cosTheta * cosTheta), Float32(Float32(alpha * alpha) - Float32(1.0)), Float32(1.0))
	return Float32(Float32(Float32(Float32(Float32(alpha * alpha) / Float32(pi)) / t_0) / t_1) - Float32(Float32(Float32(Float32(1.0) / Float32(pi)) / t_0) / 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 98.5%

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

Alternative 4: 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 98.5%

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

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 98.4%

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

Alternative 5: 97.9% accurate, N/A× speedup

Math

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

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 (* -0.5 (* (/ 1.0 t_0) t_2)))
        (t_4
         (-
          (* (/ 0.5 t_0) t_2)
          (* (/ -0.5 t_0) (/ (* cosTheta cosTheta) (pow t_1 2.0)))))
        (t_5
         (*
          (fma (/ (* (pow (* alpha cosTheta) 2.0) t_4) t_1) -1.0 t_4)
          (* alpha alpha))))
   (/ (- (* t_5 t_5) (* t_3 t_3)) (- t_5 t_3))))

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

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(Float32(-0.5) * Float32(Float32(Float32(1.0) / t_0) * t_2))
	t_4 = Float32(Float32(Float32(Float32(0.5) / t_0) * t_2) - Float32(Float32(Float32(-0.5) / t_0) * Float32(Float32(cosTheta * cosTheta) / (t_1 ^ Float32(2.0)))))
	t_5 = Float32(fma(Float32(Float32((Float32(alpha * cosTheta) ^ Float32(2.0)) * t_4) / t_1), Float32(-1.0), t_4) * Float32(alpha * alpha))
	return Float32(Float32(Float32(t_5 * t_5) - Float32(t_3 * t_3)) / Float32(t_5 - t_3))
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

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 98.4%

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

7. Applied rewrites 98.1%

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

Reproduce

herbie shell --seed 2025065 
(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)))))