GTR1 distribution

Percentage Accurate: 98.6% → 98.5%

Time: 10.0s

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: 98.6% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(pi)) / Float32(Float32(2.0) * log(alpha))) * Float32(Float32(1.0) / fma(fma(alpha, alpha, Float32(-1.0)), (cosTheta ^ Float32(2.0)), Float32(1.0))))
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. Step-by-step derivation

   1. associate-/r* 98.4%

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

   2. cancel-sign-sub 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. unsub-neg 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. associate-/r* 98.4%

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

   7. fma-neg 98.4%

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

   8. metadata-eval 98.4%

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

   9. associate-*l* 98.4%

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

   10. +-commutative 98.4%

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

3. Simplified 98.4%

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

   1. associate-/r* 98.5%

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

   2. fma-undefine 98.4%

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

   3. associate-*l* 98.4%

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

   4. *-commutative 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-/r* 98.5%

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

   7. associate-/r* 98.4%

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

   8. div-inv 98.4%

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

   9. associate-/r* 98.5%

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

   10. pow2 98.5%

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

   11. log-pow 98.5%

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

   12. +-commutative 98.5%

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

   13. *-commutative 98.5%

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

   14. associate-*l* 98.5%

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

   15. fma-undefine 98.5%

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

6. Applied egg-rr 98.5%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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(cosTheta * Float32(Float32(cosTheta * (alpha ^ Float32(2.0))) - cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) + (alpha * alpha)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) + (cosTheta * ((cosTheta * (alpha ^ single(2.0))) - 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. fma-neg 98.4%

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

   2. metadata-eval 98.4%

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

   3. *-commutative 98.4%

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

   4. fma-undefine 98.4%

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

   5. distribute-rgt-in 98.4%

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

   6. pow2 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = single(-1.0) + (alpha * alpha);
	tmp = t_0 / ((single(pi) * log((alpha * alpha))) * (single(1.0) + (cosTheta * (cosTheta * t_0))));
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. Final simplification 98.4%

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

Alternative 4: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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(cosTheta * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) + (alpha * alpha)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) - (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. Taylor expanded in alpha around 0 97.8%

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

5. Simplified 97.8%

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

6. Final simplification 97.8%

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

Alternative 5: 95.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) + (alpha * alpha)) / (single(pi) * log((alpha * alpha)));
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. Taylor expanded in alpha around 0 97.8%

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

5. Simplified 97.8%

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

6. Taylor expanded in cosTheta around 0 96.0%

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

7. Final simplification 96.0%

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

Alternative 6: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(-1.0) + (alpha * alpha)) / (log(alpha) * (single(pi) * single(2.0)));
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. Taylor expanded in alpha around 0 97.8%

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

5. Simplified 97.8%

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

6. Taylor expanded in cosTheta around 0 96.0%

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

7. Taylor expanded in alpha around 0 96.0%

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

   1. associate-*r* 96.0%

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

9. Simplified 96.0%

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

10. Final simplification 96.0%

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

Alternative 7: 65.4% accurate, 1.2× 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 98.4%

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

3. Taylor expanded in alpha around 0 97.8%

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

5. Simplified 97.8%

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

6. Taylor expanded in cosTheta around 0 96.0%

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

7. Taylor expanded in alpha around 0 67.5%

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

   1. associate-/r* 67.6%

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

9. Simplified 67.6%

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

Alternative 8: 65.4% accurate, 1.2× 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 98.4%

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

3. Taylor expanded in alpha around 0 97.8%

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

5. Simplified 97.8%

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

6. Taylor expanded in cosTheta around 0 96.0%

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

7. Taylor expanded in alpha around 0 67.5%

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

Reproduce

herbie shell --seed 2024144 
(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)))))