GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 41.9s

Alternatives: 8

Speedup: 0.8×

Specification

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (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

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (let* ((t_0 (- (* alpha alpha) 1.0)))
  (/
   t_0
   (*
    (* PI (log (* alpha alpha)))
    (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

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

Julia

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

MATLAB

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

Alternative 1: 98.6% accurate, 0.7× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (let* ((t_0 (- (* alpha alpha) 1.0)))
  (/
   t_0
   (*
    (log (pow (* alpha alpha) PI))
    (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) - 1.0f;
	return t_0 / (logf(powf((alpha * alpha), ((float) M_PI))) * (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(log((Float32(alpha * alpha) ^ Float32(pi))) * 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 / (log(((alpha * alpha) ^ single(pi))) * (single(1.0) + ((t_0 * cosTheta) * cosTheta)));
end

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.6%

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

Alternative 2: 98.5% accurate, 0.8× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (let* ((t_0 (log (* alpha alpha))))
  (/
   (- (* alpha alpha) 1.0)
   (fma
    PI
    t_0
    (* (* (* (* (fma alpha alpha -1.0) cosTheta) cosTheta) t_0) PI)))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

Alternative 3: 98.4% accurate, 1.3× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 (/ 0.5 (log alpha))
 (fma cosTheta (* cosTheta PI) (/ PI (fma alpha alpha -1.0)))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 97.9%

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

9. Applied rewrites 98.4%

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

Alternative 4: 97.7% accurate, 1.3× speedup

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

Math

\[\frac{0.31830987334251404}{\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(cosTheta, cosTheta, \frac{1}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}\right)} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/
 0.31830987334251404
 (*
  (log (* alpha alpha))
  (fma cosTheta cosTheta (/ 1.0 (fma alpha alpha -1.0))))))

C

float code(float cosTheta, float alpha) {
	return 0.31830987334251404f / (logf((alpha * alpha)) * fmaf(cosTheta, cosTheta, (1.0f / fmaf(alpha, alpha, -1.0f))));
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(0.31830987334251404) / Float32(log(Float32(alpha * alpha)) * fma(cosTheta, cosTheta, Float32(Float32(1.0) / fma(alpha, alpha, Float32(-1.0))))))
end

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 97.7%

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

8. Evaluated real constant 97.7%

   \[\leadsto \frac{0.31830987334251404}{\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{fma}\left(cosTheta, cosTheta, \frac{1}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}\right)} \]
9. Add Preprocessing

Alternative 5: 95.2% accurate, 1.8× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (fma alpha alpha -1.0) (* (* 2.0 (log alpha)) PI)))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.2%

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

8. Applied rewrites 95.1%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 95.2%

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

Alternative 6: 65.5% accurate, 2.2× speedup

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

Math

\[\frac{\frac{-1}{\pi}}{2 \cdot \log \alpha} \]

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (/ -1.0 PI) (* 2.0 (log alpha))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.0%

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

8. Applied rewrites 95.1%

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

9. Taylor expanded in alpha around 0

   \[\leadsto \frac{\frac{-1}{\pi}}{2 \cdot \log \alpha} \]

11. Applied rewrites 65.5%

    \[\leadsto \frac{\frac{-1}{\pi}}{2 \cdot \log \alpha} \]
12. Add Preprocessing

Alternative 7: 65.5% accurate, 2.2× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (/ -1.0 PI) (log (* alpha alpha))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.0%

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

8. Applied rewrites 95.0%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 65.5%

    \[\leadsto \frac{\frac{-1}{\pi}}{\log \left(\alpha \cdot \alpha\right)} \]
12. Add Preprocessing

Alternative 8: 65.5% accurate, 2.4× speedup

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

Math

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

FPCore

(FPCore (cosTheta alpha)
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0))
     (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ -1.0 (* (* 2.0 (log alpha)) PI)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 95.2%

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

7. Taylor expanded in alpha around 0

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

9. Applied rewrites 95.1%

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

10. Taylor expanded in alpha around 0

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

12. Applied rewrites 65.5%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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)))))