GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 13.5s

Alternatives: 6

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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* 98.6%

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

   2. cancel-sign-sub 98.6%

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

   3. distribute-rgt-neg-out 98.6%

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

   4. unsub-neg 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. associate-/r* 98.6%

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

   7. fma-neg 98.5%

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

   8. metadata-eval 98.5%

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

   9. associate-*l* 98.5%

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

   10. *-commutative 98.5%

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

   11. distribute-rgt-neg-out 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in alpha around 0 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 2: 98.5% 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 98.6%

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

3. Final simplification 98.6%

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

Alternative 3: 97.4% 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 98.6%

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

3. Taylor expanded in alpha around 0 97.9%

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

5. Simplified 97.9%

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

6. Final simplification 97.9%

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

Alternative 4: 94.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\left(\alpha + 1\right) \cdot \left(\alpha + -1\right)}{\pi \cdot \log \alpha} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* 98.6%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. +-commutative 98.5%

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

   5. associate-*l* 98.5%

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

   6. fma-define 98.5%

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

   7. fma-neg 98.5%

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

   8. metadata-eval 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.3%

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

   2. fma-undefine 98.4%

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

   3. difference-of-sqr--1 98.0%

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

   4. associate-*l* 97.9%

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

   5. sub-neg 97.9%

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

   6. metadata-eval 97.9%

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

   7. associate-/r* 97.7%

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

   8. pow2 97.7%

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

   9. log-pow 97.9%

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

6. Applied egg-rr 97.9%

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

7. Taylor expanded in cosTheta around 0 95.9%

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

8. Final simplification 95.9%

   \[\leadsto 0.5 \cdot \frac{\left(\alpha + 1\right) \cdot \left(\alpha + -1\right)}{\pi \cdot \log \alpha} \]
9. Add Preprocessing

Alternative 5: 65.1% 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)) / Float32(-log(alpha)))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* 98.6%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. +-commutative 98.5%

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

   5. associate-*l* 98.5%

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

   6. fma-define 98.5%

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

   7. fma-neg 98.5%

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

   8. metadata-eval 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in alpha around 0 68.9%

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

6. Taylor expanded in cosTheta around 0 67.9%

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

7. Taylor expanded in alpha around inf 67.9%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{\pi \cdot \log \left(\frac{1}{\alpha}\right)}}}{1} \]
8. Step-by-step derivation

   1. *-un-lft-identity 67.9%

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

   2. neg-log 67.9%

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

   3. expm1-log1p-u 67.9%

      \[\leadsto \frac{1 \cdot \frac{0.5}{\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\log \alpha\right)\right)}}}{1} \]

   4. associate-/r* 67.9%

      \[\leadsto \frac{1 \cdot \color{blue}{\frac{\frac{0.5}{\pi}}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\log \alpha\right)\right)}}}{1} \]

   5. expm1-log1p-u 67.9%

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

9. Applied egg-rr 67.9%

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

10. Final simplification 67.9%

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

Alternative 6: 65.1% 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 98.6%

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

   1. associate-/r* 98.6%

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

   2. fma-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. +-commutative 98.5%

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

   5. associate-*l* 98.5%

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

   6. fma-define 98.5%

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

   7. fma-neg 98.5%

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

   8. metadata-eval 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in alpha around 0 68.9%

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

6. Taylor expanded in cosTheta around 0 67.9%

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

7. Final simplification 67.9%

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

Reproduce

herbie shell --seed 2024088 
(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)))))