GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 12.5s

Alternatives: 8

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. fma-neg 98.7%

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

   2. metadata-eval 98.7%

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

   3. *-commutative 98.7%

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

   4. fma-udef 98.7%

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

   5. distribute-rgt-in 98.7%

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

   6. pow2 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (* PI (log (* alpha alpha))) (+ 1.0 (* cosTheta (* t_0 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 + (cosTheta * (t_0 * 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(cosTheta * Float32(t_0 * 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) + (cosTheta * (t_0 * cosTheta))));
end

Derivation

1. Initial program 98.7%

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

2. Final simplification 98.7%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in alpha around 0 97.6%

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

   1. mul-1-neg 97.6%

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

4. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 4: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in alpha around 0 97.6%

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

   1. mul-1-neg 97.6%

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

4. Simplified 97.6%

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

5. Taylor expanded in alpha around 0 67.9%

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

6. Taylor expanded in alpha around inf 67.9%

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

7. Final simplification 67.9%

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

Alternative 5: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in alpha around 0 97.6%

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

   1. mul-1-neg 97.6%

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

4. Simplified 97.6%

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

5. Taylor expanded in alpha around 0 67.9%

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

6. Final simplification 67.9%

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

Alternative 6: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\log \alpha} \cdot \frac{-1}{\pi} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (* (/ 0.5 (log alpha)) (/ -1.0 PI)))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(0.5) / log(alpha)) * (single(-1.0) / single(pi));
end

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in cosTheta around 0 94.1%

   \[\leadsto \color{blue}{\frac{{\alpha}^{2} - 1}{\pi \cdot \log \left({\alpha}^{2}\right)}} \]
3. Step-by-step derivation

   1. unpow2 94.1%

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

   2. fma-neg 93.9%

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

   3. metadata-eval 93.9%

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

   4. *-lft-identity 93.9%

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

   5. log-pow 94.0%

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

   6. *-commutative 94.0%

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

   7. *-commutative 94.0%

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

   8. count-2 94.0%

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

   9. distribute-lft-out 94.0%

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

   10. count-2 94.0%

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

   11. times-frac 94.0%

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

   12. metadata-eval 94.0%

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

   13. associate-*r/ 94.0%

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

   14. *-commutative 94.0%

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

   15. times-frac 93.8%

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

4. Simplified 93.8%

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

5. Taylor expanded in alpha around 0 65.8%

   \[\leadsto \frac{0.5}{\log \alpha} \cdot \color{blue}{\frac{-1}{\pi}} \]

6. Final simplification 65.8%

   \[\leadsto \frac{0.5}{\log \alpha} \cdot \frac{-1}{\pi} \]

Alternative 7: 66.2% accurate, 1.1× speedup

Math

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

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(-0.5) / Float32(Float32(pi) * log(alpha)))
end

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in cosTheta around 0 94.1%

   \[\leadsto \color{blue}{\frac{{\alpha}^{2} - 1}{\pi \cdot \log \left({\alpha}^{2}\right)}} \]
3. Step-by-step derivation

   1. unpow2 94.1%

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

   2. fma-neg 93.9%

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

   3. metadata-eval 93.9%

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

   4. *-lft-identity 93.9%

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

   5. log-pow 94.0%

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

   6. *-commutative 94.0%

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

   7. *-commutative 94.0%

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

   8. count-2 94.0%

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

   9. distribute-lft-out 94.0%

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

   10. count-2 94.0%

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

   11. times-frac 94.0%

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

   12. metadata-eval 94.0%

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

   13. associate-*r/ 94.0%

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

   14. *-commutative 94.0%

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

   15. times-frac 93.8%

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

4. Simplified 93.8%

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

5. Taylor expanded in alpha around 0 65.8%

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

6. Final simplification 65.8%

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

Alternative 8: 66.2% accurate, 1.1× 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 98.7%

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

2. Taylor expanded in cosTheta around 0 94.1%

   \[\leadsto \color{blue}{\frac{{\alpha}^{2} - 1}{\pi \cdot \log \left({\alpha}^{2}\right)}} \]
3. Step-by-step derivation

   1. unpow2 94.1%

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

   2. fma-neg 93.9%

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

   3. metadata-eval 93.9%

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

   4. *-lft-identity 93.9%

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

   5. log-pow 94.0%

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

   6. *-commutative 94.0%

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

   7. *-commutative 94.0%

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

   8. count-2 94.0%

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

   9. distribute-lft-out 94.0%

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

   10. count-2 94.0%

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

   11. times-frac 94.0%

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

   12. metadata-eval 94.0%

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

   13. associate-*r/ 94.0%

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

   14. *-commutative 94.0%

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

   15. times-frac 93.8%

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

4. Simplified 93.8%

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

5. Taylor expanded in alpha around 0 65.8%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \log \alpha}} \]
6. Step-by-step derivation

   1. associate-/r* 65.8%

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

7. Simplified 65.8%

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

8. Final simplification 65.8%

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

Reproduce

herbie shell --seed 2023300 
(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)))))