GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 11.0s

Alternatives: 10

Speedup: 0.6×

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: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. clear-num N/A

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

   5. flip-+ N/A

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

   6. lift-+.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 98.5

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l* N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in alpha around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unpow2 N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 N/A

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in alpha around 0

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

   1. sub-neg N/A

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

   2. unpow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 98.7

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

10. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f32 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. clear-num N/A

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

   5. flip-+ N/A

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

   6. lift-+.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 98.5

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l* N/A

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024230 
(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)))))