GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 10.5s

Alternatives: 11

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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, 0.7× 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. 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%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

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, (((float) M_PI) * 2.0f))));
}

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(pi) * Float32(2.0))))))
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(pi) * single(2.0)))));
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. 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%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 98.4%

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

   1. associate-*r* 98.4%

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

   2. log-pow 98.5%

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

   3. log-pow 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l* 98.4%

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

   6. exp-to-pow 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((1.0f + (((alpha * alpha) * (cosTheta * cosTheta)) + -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(Float32(Float32(1.0) + Float32(Float32(Float32(alpha * alpha) * Float32(cosTheta * cosTheta)) + 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) + ((single(1.0) + (((alpha * alpha) * (cosTheta * cosTheta)) + 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. Step-by-step derivation

   1. fma-neg 98.6%

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

   2. metadata-eval 98.6%

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

   3. associate-*r* 98.6%

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

   4. *-commutative 98.6%

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

   5. fma-udef 98.6%

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

   6. distribute-lft-in 98.6%

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

3. Applied egg-rr 98.6%

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

   1. expm1-log1p-u 98.6%

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

   2. expm1-udef 98.5%

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

   3. log1p-udef 98.5%

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

   4. add-exp-log 98.5%

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

   5. *-commutative 98.5%

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

5. Applied egg-rr 98.5%

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

   1. associate--l+ 98.6%

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

   2. unswap-sqr 98.6%

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

   3. *-commutative 98.6%

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

   4. *-commutative 98.6%

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

   5. swap-sqr 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) + -1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + (((alpha * alpha) * (cosTheta * cosTheta)) - (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(Float32(Float32(alpha * alpha) * Float32(cosTheta * cosTheta)) - Float32(cosTheta * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) + single(-1.0)) / ((single(pi) * log((alpha * alpha))) * (single(1.0) + (((alpha * alpha) * (cosTheta * cosTheta)) - (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. Step-by-step derivation

   1. fma-neg 98.6%

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

   2. metadata-eval 98.6%

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

   3. associate-*r* 98.6%

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

   4. *-commutative 98.6%

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

   5. fma-udef 98.6%

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

   6. distribute-lft-in 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 5: 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 98.6%

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

2. Final simplification 98.6%

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

Alternative 6: 97.4% accurate, 1.0× speedup

Math

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

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(2.0) * Float32(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 98.6%

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

2. Taylor expanded in alpha around inf 98.2%

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

3. Taylor expanded in alpha around 0 97.5%

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

5. Simplified 97.5%

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

6. Taylor expanded in alpha around 0 97.7%

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

7. Final simplification 97.7%

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

Alternative 7: 97.4% accurate, 1.0× 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. Taylor expanded in alpha around 0 97.8%

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

4. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 8: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* 98.6%

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

   2. difference-of-sqr-1 98.2%

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

   3. *-commutative 98.2%

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

   4. times-frac 98.0%

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

   5. *-commutative 98.0%

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

   6. times-frac 98.2%

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

   7. difference-of-sqr-1 98.6%

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

   8. associate-/l/ 98.6%

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

   9. log-prod 98.6%

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

   10. count-2 98.6%

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

   11. *-commutative 98.6%

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

   12. fma-neg 98.6%

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

   13. metadata-eval 98.6%

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

   14. +-commutative 98.6%

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

3. Simplified 98.7%

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

   1. associate-/l/ 98.6%

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

   2. fma-udef 98.5%

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

   3. difference-of-sqr--1 98.3%

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

   4. times-frac 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

5. Applied egg-rr 98.2%

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

6. Taylor expanded in cosTheta around 0 95.8%

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

   1. associate-/l* 95.6%

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

   2. +-commutative 95.6%

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

   3. associate-/l* 95.8%

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

   4. *-commutative 95.8%

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

   5. sub-neg 95.8%

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

   6. metadata-eval 95.8%

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

8. Simplified 95.8%

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

9. Final simplification 95.8%

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

Alternative 9: 66.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in alpha around 0 68.2%

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

   1. associate-/r* 68.2%

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

   2. *-commutative 68.2%

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

   3. mul-1-neg 68.2%

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

   4. unsub-neg 68.2%

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

   5. unpow2 68.2%

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

6. Simplified 68.2%

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

7. Final simplification 68.2%

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

Alternative 10: 66.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(-0.5) / (log(alpha) * (single(pi) * (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. Taylor expanded in alpha around 0 68.2%

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

4. Simplified 68.2%

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

5. Final simplification 68.2%

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

Alternative 11: 65.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* 98.6%

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

   2. difference-of-sqr-1 98.2%

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

   3. *-commutative 98.2%

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

   4. times-frac 98.0%

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

   5. *-commutative 98.0%

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

   6. times-frac 98.2%

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

   7. difference-of-sqr-1 98.6%

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

   8. associate-/l/ 98.6%

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

   9. log-prod 98.6%

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

   10. count-2 98.6%

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

   11. *-commutative 98.6%

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

   12. fma-neg 98.6%

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

   13. metadata-eval 98.6%

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

   14. +-commutative 98.6%

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

3. Simplified 98.7%

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

4. Taylor expanded in alpha around 0 68.3%

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

6. Simplified 68.3%

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

7. Taylor expanded in cosTheta around 0 67.4%

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

8. Taylor expanded in alpha around 0 67.4%

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

9. Final simplification 67.4%

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

Reproduce

herbie shell --seed 2023174 
(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)))))