GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 28.1s

Alternatives: 9

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: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t_0}{\log \left({\left({\alpha}^{2}\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (log (pow (pow alpha 2.0) PI)) (+ 1.0 (* cosTheta (* t_0 cosTheta)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) + -1.0f;
	return t_0 / (logf(powf(powf(alpha, 2.0f), ((float) M_PI))) * (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(log(((alpha ^ Float32(2.0)) ^ Float32(pi))) * 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 / (log(((alpha ^ single(2.0)) ^ single(pi))) * (single(1.0) + (cosTheta * (t_0 * 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
3. Step-by-step derivation

   1. add-log-exp 98.4%

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

   2. *-commutative 98.4%

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

   3. exp-to-pow 98.8%

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

   4. pow2 98.8%

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right) \cdot \log \left({\alpha}^{\left(2 \cdot \pi\right)}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (+ 1.0 (* cosTheta (* t_0 cosTheta))) (log (pow alpha (* 2.0 PI)))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) + -1.0f;
	return t_0 / ((1.0f + (cosTheta * (t_0 * cosTheta))) * logf(powf(alpha, (2.0f * ((float) M_PI)))));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) + Float32(-1.0))
	return Float32(t_0 / Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(t_0 * cosTheta))) * log((alpha ^ Float32(Float32(2.0) * Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) + single(-1.0);
	tmp = t_0 / ((single(1.0) + (cosTheta * (t_0 * cosTheta))) * log((alpha ^ (single(2.0) * 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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 98.4%

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

   2. *-commutative 98.4%

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

   3. exp-to-pow 98.8%

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

   4. pow2 98.8%

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in alpha around 0 98.5%

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

   1. associate-*r* 98.5%

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

   2. *-commutative 98.5%

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

   3. exp-to-pow 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) + single(-1.0);
	tmp = t_0 / ((single(1.0) + (cosTheta * (t_0 * cosTheta))) * (single(pi) * log((alpha * 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. Add Preprocessing

3. Final simplification 98.5%

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

Alternative 4: 97.5% accurate, 1.1× 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 98.5%

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

3. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

5. Simplified 97.3%

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

6. Final simplification 97.3%

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

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / ((single(1.0) - (cosTheta * cosTheta)) * ((single(2.0) * 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. Add Preprocessing

3. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

5. Simplified 97.3%

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

6. Taylor expanded in alpha around 0 97.3%

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

   1. count-2 97.3%

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

   2. distribute-rgt-out 97.3%

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

   3. count-2 97.3%

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

8. Simplified 97.3%

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

9. Final simplification 97.3%

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

Alternative 6: 94.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (alpha + single(1.0)) / (((single(2.0) * 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. Add Preprocessing

3. Taylor expanded in alpha around 0 97.3%

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

   1. mul-1-neg 97.3%

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

5. Simplified 97.3%

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

   1. difference-of-sqr-1 97.1%

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

   2. *-un-lft-identity 97.1%

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

   3. fma-neg 97.1%

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

   4. metadata-eval 97.1%

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

   5. fma-def 97.1%

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

   6. *-un-lft-identity 97.1%

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

7. Applied egg-rr 97.1%

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

8. Taylor expanded in cosTheta around 0 93.3%

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

   1. sub-neg 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-/l* 93.1%

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

   4. +-commutative 93.1%

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

   5. log-pow 93.1%

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

   6. associate-*r* 93.1%

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

   7. *-commutative 93.1%

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

10. Simplified 93.1%

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

11. Final simplification 93.1%

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

Alternative 7: 66.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Taylor expanded in alpha around 0 69.2%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]
4. Step-by-step derivation

   1. mul-1-neg 69.2%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)\right)} \]

5. Simplified 69.2%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \left(-{cosTheta}^{2}\right)\right)\right)}} \]
6. Step-by-step derivation

   1. unpow2 69.2%

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

7. Applied egg-rr 69.2%

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

8. Final simplification 69.2%

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

Alternative 8: 65.2% 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 98.5%

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

3. Taylor expanded in alpha around 0 69.2%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]
4. Step-by-step derivation

   1. mul-1-neg 69.2%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)\right)} \]

5. Simplified 69.2%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \left(-{cosTheta}^{2}\right)\right)\right)}} \]

6. Taylor expanded in cosTheta around 0 66.5%

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

7. Final simplification 66.5%

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

Alternative 9: 65.2% 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 98.5%

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

3. Taylor expanded in alpha around 0 69.2%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]
4. Step-by-step derivation

   1. mul-1-neg 69.2%

      \[\leadsto \frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \color{blue}{\left(-{cosTheta}^{2}\right)}\right)\right)} \]

5. Simplified 69.2%

   \[\leadsto \color{blue}{\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \left(-{cosTheta}^{2}\right)\right)\right)}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 69.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \left(-{cosTheta}^{2}\right)\right)\right)}\right)\right)} \]

   2. expm1-udef 67.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\pi \cdot \left(\log \alpha \cdot \left(1 + \left(-{cosTheta}^{2}\right)\right)\right)}\right)} - 1} \]

7. Applied egg-rr 64.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\pi \cdot \left(\mathsf{fma}\left(cosTheta, cosTheta, 1\right) \cdot \log \alpha\right)}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 66.2%

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

   2. expm1-log1p 66.2%

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

   3. associate-/r* 66.2%

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

   4. *-commutative 66.2%

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

9. Simplified 66.2%

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

10. Taylor expanded in cosTheta around 0 66.5%

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

    1. associate-/r* 66.5%

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

12. Simplified 66.5%

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

13. Final simplification 66.5%

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

Reproduce

herbie shell --seed 2024017 
(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)))))