GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 19.3s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. associate-/r* 98.6%

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

   2. cancel-sign-sub 98.6%

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

   3. distribute-rgt-neg-out 98.6%

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

   4. unsub-neg 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. associate-/r* 98.6%

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

   7. sqr-neg 98.6%

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

   8. sqr-neg 98.6%

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

   9. fma-neg 98.5%

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

   10. metadata-eval 98.5%

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

   11. associate-*l* 98.6%

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

3. Simplified 98.7%

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

5. Taylor expanded in cosTheta around 0 98.7%

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

   1. distribute-lft-out 98.7%

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

   2. unpow2 98.7%

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

   3. fma-neg 98.7%

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

   4. metadata-eval 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(Float32(pi) * Float32(Float32(Float32(2.0) * log(alpha)) * fma(fma(alpha, alpha, Float32(-1.0)), Float32(cosTheta * cosTheta), Float32(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. Step-by-step derivation

   1. associate-/r* 98.6%

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

   2. cancel-sign-sub 98.6%

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

   3. distribute-rgt-neg-out 98.6%

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

   4. unsub-neg 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. associate-/r* 98.6%

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

   7. sqr-neg 98.6%

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

   8. sqr-neg 98.6%

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

   9. fma-neg 98.5%

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

   10. metadata-eval 98.5%

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

   11. associate-*l* 98.6%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha + -1\\ \frac{t\_0}{\log \left({\left({\alpha}^{2}\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot t\_0\right)\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (alpha * alpha) + single(-1.0);
	tmp = t_0 / (log(((alpha ^ single(2.0)) ^ single(pi))) * (single(1.0) + (cosTheta * (cosTheta * t_0))));
end

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. add-log-exp 98.6%

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

   3. exp-to-pow 98.6%

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

   4. pow2 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta, float alpha) {
	float t_0 = (alpha * alpha) + -1.0f;
	return t_0 / ((1.0f + (cosTheta * (cosTheta * t_0))) * (((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(cosTheta * t_0))) * 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 * (cosTheta * t_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. Final simplification 98.6%

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

Alternative 5: 97.6% 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 98.0%

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

   1. mul-1-neg 98.0%

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

5. Simplified 98.0%

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

6. Final simplification 98.0%

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

Alternative 6: 95.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = single(1.0) / (single(2.0) * ((single(pi) / (alpha + single(1.0))) * (log(alpha) / (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 98.0%

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

   1. mul-1-neg 98.0%

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

5. Simplified 98.0%

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

   1. fma-neg 97.9%

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

   2. metadata-eval 97.9%

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

   3. clear-num 97.8%

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

   4. inv-pow 97.8%

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

7. Applied egg-rr 95.7%

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

   1. unpow-1 95.7%

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

   2. associate-*r* 95.7%

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

   3. associate-*r* 95.7%

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

   4. *-commutative 95.7%

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

   5. associate-*r* 95.7%

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

9. Simplified 95.7%

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

10. Taylor expanded in cosTheta around 0 95.9%

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

    1. pow2 95.9%

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

    2. difference-of-sqr-1 95.6%

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

    3. add-exp-log 95.6%

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

    4. expm1-udef 95.6%

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

    5. times-frac 95.8%

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

    6. expm1-udef 95.8%

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

    7. add-exp-log 95.8%

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

12. Applied egg-rr 95.8%

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

13. Final simplification 95.8%

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

Alternative 7: 95.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha + single(1.0)) * ((alpha + single(-1.0)) / single(pi))) * (single(0.5) / 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 98.0%

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

   1. mul-1-neg 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in cosTheta around 0 96.1%

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

   1. unpow2 96.1%

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

   2. fma-neg 96.0%

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

   3. metadata-eval 96.0%

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

   4. *-rgt-identity 96.0%

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

   5. log-pow 96.1%

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

   6. /-rgt-identity 96.1%

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

   7. times-frac 95.7%

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

   8. /-rgt-identity 95.7%

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

   9. associate-/r* 95.7%

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

   10. metadata-eval 95.7%

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

8. Simplified 95.7%

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

   1. metadata-eval 95.7%

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

   2. fma-neg 95.7%

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

   3. difference-of-sqr-1 95.5%

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

   4. add-exp-log 95.5%

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

   5. expm1-udef 95.5%

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

   6. *-un-lft-identity 95.5%

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

   7. times-frac 95.5%

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

   8. expm1-udef 95.5%

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

   9. add-exp-log 95.4%

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

10. Applied egg-rr 95.4%

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

11. Final simplification 95.4%

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

Alternative 8: 67.0% accurate, 1.1× speedup

Math

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

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(Float32(-0.5) / log(alpha)) / 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. 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. sqr-neg 98.6%

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

   3. sqr-neg 98.6%

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

   4. fma-neg 98.6%

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

   5. metadata-eval 98.6%

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

   6. *-commutative 98.6%

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

   7. cancel-sign-sub 98.6%

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

   8. cancel-sign-sub-inv 98.6%

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

   9. remove-double-neg 98.6%

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

   10. sqr-neg 98.6%

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

   11. sqr-neg 98.6%

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

   12. fma-neg 98.6%

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

   13. metadata-eval 98.6%

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

3. Simplified 98.6%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.2%

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

   3. pow2 98.2%

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

   4. log-pow 98.3%

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

   5. *-commutative 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in alpha around 0 97.7%

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

   1. mul-1-neg 97.7%

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

9. Simplified 97.7%

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

    1. associate-*l/ 97.9%

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

    2. *-commutative 97.9%

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

11. Applied egg-rr 97.9%

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

12. Taylor expanded in alpha around 0 68.4%

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

13. Final simplification 68.4%

    \[\leadsto \frac{\frac{\frac{-0.5}{\log \alpha}}{\pi}}{1 - cosTheta \cdot cosTheta} \]
14. Add Preprocessing

Alternative 9: 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 98.6%

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

3. Taylor expanded in alpha around 0 68.3%

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

   1. associate-*r* 68.3%

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

   2. mul-1-neg 68.3%

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

   3. unsub-neg 68.3%

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

5. Simplified 68.3%

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

6. Taylor expanded in cosTheta around 0 67.4%

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

7. Final simplification 67.4%

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

Alternative 10: 65.9% 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 98.0%

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

   1. mul-1-neg 98.0%

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

5. Simplified 98.0%

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

6. Taylor expanded in cosTheta around 0 96.1%

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

   1. unpow2 96.1%

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

   2. fma-neg 96.0%

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

   3. metadata-eval 96.0%

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

   4. *-rgt-identity 96.0%

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

   5. log-pow 96.1%

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

   6. /-rgt-identity 96.1%

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

   7. times-frac 95.7%

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

   8. /-rgt-identity 95.7%

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

   9. associate-/r* 95.7%

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

   10. metadata-eval 95.7%

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

8. Simplified 95.7%

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

9. Taylor expanded in alpha around 0 67.4%

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

    1. associate-/r* 67.5%

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

11. Simplified 67.5%

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

12. Final simplification 67.5%

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

Reproduce

herbie shell --seed 2024026 
(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)))))