GTR1 distribution

Percentage Accurate: 98.6% → 98.7%

Time: 10.6s

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: 98.6% 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: 98.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (+ (* alpha alpha) -1.0)))
   (/
    t_0
    (* (log (pow (* alpha alpha) 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((alpha * alpha), ((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((Float32(alpha * alpha) ^ 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 * alpha) ^ single(pi))) * (single(1.0) + (cosTheta * (t_0 * 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. Step-by-step derivation

   1. add-log-exp 98.5%

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

   2. *-commutative 98.5%

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

   3. exp-to-pow 98.7%

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

   4. pow2 98.7%

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

4. Applied egg-rr 98.7%

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

   1. pow2 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 2: 98.6% 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 98.6%

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

3. Final simplification 98.6%

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

Alternative 3: 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 98.6%

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

3. Taylor expanded in alpha around 0 98.3%

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

5. Simplified 98.3%

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

6. Final simplification 98.3%

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

Alternative 4: 95.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha + single(-1.0)) / (single(pi) * (log(alpha) * single(2.0)))) * (alpha + single(1.0));
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. Taylor expanded in alpha around 0 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in cosTheta around 0 96.3%

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

   1. difference-of-sqr-1 96.0%

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

   2. *-un-lft-identity 96.0%

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

   3. fma-neg 96.0%

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

   4. metadata-eval 96.0%

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

   5. fma-define 96.0%

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

   6. *-un-lft-identity 96.0%

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

8. Applied egg-rr 96.0%

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

   1. associate-/l* 96.1%

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

   2. log-pow 96.1%

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

   3. *-rgt-identity 96.1%

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

   4. log-pow 96.1%

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

   5. pow2 96.1%

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

   6. pow-to-exp 96.1%

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

   7. rem-log-exp 96.2%

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

10. Applied egg-rr 96.2%

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

11. Final simplification 96.2%

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

Alternative 5: 95.0% 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. Add Preprocessing

3. Taylor expanded in alpha around 0 98.3%

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

5. Simplified 98.3%

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

   1. clear-num 98.3%

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

   2. inv-pow 98.3%

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

   3. *-commutative 98.3%

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

   4. +-commutative 98.3%

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

   5. fma-define 98.3%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 96.2%

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

   8. sqr-neg 96.2%

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

   9. sqrt-prod 96.2%

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

   10. add-sqr-sqrt 96.2%

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

   11. pow2 96.2%

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

   12. log-pow 96.1%

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

   13. fma-neg 96.2%

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

   14. metadata-eval 96.2%

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

7. Applied egg-rr 96.2%

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

   1. unpow-1 96.2%

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

   2. associate-/l* 96.2%

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

   3. *-commutative 96.2%

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

   4. *-commutative 96.2%

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

   5. associate-*r* 96.2%

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

9. Simplified 96.2%

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

10. Taylor expanded in cosTheta around 0 96.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{\alpha}^{2} - 1}{\pi \cdot \log \alpha}} \]
11. Step-by-step derivation

    1. associate-/r* 96.2%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{{\alpha}^{2} - 1}{\pi}}{\log \alpha}} \]

    2. unpow2 96.2%

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

    3. fma-neg 96.2%

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

    4. metadata-eval 96.2%

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

12. Simplified 96.2%

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

    1. metadata-eval 96.2%

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

    2. fma-neg 96.2%

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

    3. associate-/l/ 96.3%

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

    4. difference-of-sqr-1 96.1%

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

    5. *-un-lft-identity 96.1%

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

    6. fma-neg 96.1%

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

    7. metadata-eval 96.1%

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

    8. fma-define 96.1%

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

    9. *-un-lft-identity 96.1%

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

    10. *-commutative 96.1%

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

    11. *-un-lft-identity 96.1%

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

    12. times-frac 96.2%

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

14. Applied egg-rr 96.2%

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

    1. /-rgt-identity 96.2%

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

    2. associate-*r/ 96.1%

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

    3. +-commutative 96.1%

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

    4. +-commutative 96.1%

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

16. Simplified 96.1%

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

17. Final simplification 96.1%

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

Alternative 6: 94.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(0.5) * ((alpha + single(1.0)) * (((alpha + single(-1.0)) / log(alpha)) / single(pi)));
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 98.3%

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

5. Simplified 98.3%

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

   1. clear-num 98.3%

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

   2. inv-pow 98.3%

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

   3. *-commutative 98.3%

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

   4. +-commutative 98.3%

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

   5. fma-define 98.3%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 96.2%

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

   8. sqr-neg 96.2%

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

   9. sqrt-prod 96.2%

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

   10. add-sqr-sqrt 96.2%

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

   11. pow2 96.2%

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

   12. log-pow 96.1%

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

   13. fma-neg 96.2%

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

   14. metadata-eval 96.2%

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

7. Applied egg-rr 96.2%

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

   1. unpow-1 96.2%

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

   2. associate-/l* 96.2%

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

   3. *-commutative 96.2%

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

   4. *-commutative 96.2%

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

   5. associate-*r* 96.2%

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

9. Simplified 96.2%

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

10. Taylor expanded in cosTheta around 0 96.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{\alpha}^{2} - 1}{\pi \cdot \log \alpha}} \]
11. Step-by-step derivation

    1. associate-/r* 96.2%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{{\alpha}^{2} - 1}{\pi}}{\log \alpha}} \]

    2. unpow2 96.2%

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

    3. fma-neg 96.2%

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

    4. metadata-eval 96.2%

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

12. Simplified 96.2%

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

    1. metadata-eval 96.2%

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

    2. fma-neg 96.2%

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

    3. associate-/l/ 96.3%

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

    4. difference-of-sqr-1 96.1%

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

    5. *-un-lft-identity 96.1%

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

    6. fma-neg 96.1%

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

    7. metadata-eval 96.1%

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

    8. fma-define 96.1%

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

    9. *-un-lft-identity 96.1%

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

    10. *-commutative 96.1%

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

    11. *-un-lft-identity 96.1%

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

    12. times-frac 96.2%

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

14. Applied egg-rr 96.2%

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

    1. /-rgt-identity 96.2%

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

    2. +-commutative 96.2%

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

    3. *-commutative 96.2%

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

    4. associate-/r* 96.0%

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

    5. +-commutative 96.0%

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

16. Simplified 96.0%

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

17. Final simplification 96.0%

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

Alternative 7: 95.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Taylor expanded in alpha around 0 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in cosTheta around 0 96.3%

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

7. Final simplification 96.3%

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

Alternative 8: 65.6% 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 98.6%

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

3. Taylor expanded in alpha around 0 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in cosTheta around 0 96.3%

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

7. Taylor expanded in alpha around 0 66.0%

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

   1. associate-/r* 66.0%

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

9. Simplified 66.0%

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

Alternative 9: 65.5% 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. Add Preprocessing

3. Taylor expanded in alpha around 0 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in cosTheta around 0 96.3%

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

7. Taylor expanded in alpha around 0 66.0%

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

Reproduce

herbie shell --seed 2024145 
(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)))))