GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 10.6s

Alternatives: 8

Speedup: 1.0×

• Rival profile vector overflowed, profile may not be complete

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, 0.6× 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 98.4%

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

   1. *-commutative 98.4%

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

   2. add-log-exp 98.3%

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

   3. exp-to-pow 98.4%

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

   4. pow2 98.4%

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

4. Applied egg-rr 98.4%

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

   1. pow2 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 98.5% 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 98.4%

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

3. Final simplification 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha + -1}{\left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha + -1\right) \cdot cosTheta\right)\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \]
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 98.4%

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

3. Taylor expanded in alpha around 0 97.1%

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

5. Simplified 97.1%

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

6. Final simplification 97.1%

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

Alternative 4: 94.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in alpha around 0 97.1%

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

5. Simplified 97.1%

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

   1. difference-of-sqr-1 96.8%

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

   2. *-un-lft-identity 96.8%

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

   3. fmm-def 96.8%

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

   4. metadata-eval 96.8%

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

   5. fma-define 96.8%

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

   6. *-un-lft-identity 96.8%

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

   7. times-frac 96.9%

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

   8. pow2 96.9%

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

   9. log-pow 97.0%

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

   10. +-commutative 97.0%

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

   11. fma-define 97.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 93.3%

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

   14. sqr-neg 93.3%

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

   15. sqrt-prod 93.3%

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

   16. add-sqr-sqrt 93.3%

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

7. Applied egg-rr 93.3%

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

8. Taylor expanded in cosTheta around 0 93.7%

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

9. Final simplification 93.7%

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

Alternative 5: 94.9% 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 98.4%

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

3. Taylor expanded in alpha around 0 97.1%

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

5. Simplified 97.1%

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

   1. difference-of-sqr-1 96.8%

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

   2. *-un-lft-identity 96.8%

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

   3. fmm-def 96.8%

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

   4. metadata-eval 96.8%

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

   5. fma-define 96.8%

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

   6. *-un-lft-identity 96.8%

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

   7. times-frac 96.9%

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

   8. pow2 96.9%

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

   9. log-pow 97.0%

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

   10. +-commutative 97.0%

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

   11. fma-define 97.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 93.3%

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

   14. sqr-neg 93.3%

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

   15. sqrt-prod 93.3%

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

   16. add-sqr-sqrt 93.3%

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

7. Applied egg-rr 93.3%

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

8. Taylor expanded in cosTheta around 0 93.6%

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

9. Final simplification 93.6%

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

Alternative 6: 94.9% accurate, 1.1× speedup

Math

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

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(alpha + Float32(1.0)) * Float32(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 98.4%

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

3. Taylor expanded in alpha around 0 97.1%

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

5. Simplified 97.1%

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

   1. difference-of-sqr-1 96.8%

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

   2. *-un-lft-identity 96.8%

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

   3. fmm-def 96.8%

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

   4. metadata-eval 96.8%

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

   5. fma-define 96.8%

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

   6. *-un-lft-identity 96.8%

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

   7. times-frac 96.9%

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

   8. pow2 96.9%

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

   9. log-pow 97.0%

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

   10. +-commutative 97.0%

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

   11. fma-define 97.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 93.3%

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

   14. sqr-neg 93.3%

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

   15. sqrt-prod 93.3%

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

   16. add-sqr-sqrt 93.3%

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

7. Applied egg-rr 93.3%

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

8. Taylor expanded in cosTheta around 0 93.6%

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

   1. sub-neg 93.6%

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

   2. metadata-eval 93.6%

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

   3. associate-/l* 93.6%

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

   4. +-commutative 93.6%

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

10. Simplified 93.6%

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

Alternative 7: 95.2% 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 98.4%

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

3. Taylor expanded in alpha around 0 97.1%

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

5. Simplified 97.1%

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

6. Taylor expanded in cosTheta around 0 93.7%

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

7. Final simplification 93.7%

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

Alternative 8: 65.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in alpha around 0 97.1%

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

5. Simplified 97.1%

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

6. Taylor expanded in cosTheta around 0 93.7%

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

7. Taylor expanded in alpha around 0 65.6%

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

Reproduce

herbie shell --seed 2024155 
(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)))))