GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 3.8s

Alternatives: 10

Speedup: 1.0×

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 98.5%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 98.4%

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

Alternative 3: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 98.4%

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

Alternative 4: 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

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites 97.6%

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

Alternative 6: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 97.6%

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

Alternative 7: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((single(pi) * log((alpha * alpha))) * single(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. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.2%

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

Alternative 8: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 95.2%

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

Alternative 9: 65.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5}{\pi}}{\log \alpha} \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ (/ -0.5 PI) (log alpha)))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-0.5) / single(pi)) / log(alpha);
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 65.1%

   \[\leadsto \frac{-0.5}{\color{blue}{\log \alpha \cdot \pi}} \]

9. Applied rewrites 65.2%

   \[\leadsto \frac{\frac{-0.5}{\pi}}{\log \alpha} \]
10. Add Preprocessing

Alternative 10: 65.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\log \alpha \cdot \pi} \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ -0.5 (* (log alpha) PI)))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-0.5) / (log(alpha) * single(pi));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 65.1%

   \[\leadsto \frac{-0.5}{\color{blue}{\log \alpha \cdot \pi}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025100 
(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)))))