GTR1 distribution

Percentage Accurate: 98.6% → 98.7%

Time: 3.8s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{\alpha}} \cdot \alpha\\ t_1 := t\_0 - 1\\ \frac{t\_1}{\log \left({t\_0}^{\pi}\right) \cdot \left(1 + \left(t\_1 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (* (/ 1.0 (/ 1.0 alpha)) alpha)) (t_1 (- t_0 1.0)))
   (/ t_1 (* (log (pow t_0 PI)) (+ 1.0 (* (* t_1 cosTheta) cosTheta))))))

C

float code(float cosTheta, float alpha) {
	float t_0 = (1.0f / (1.0f / alpha)) * alpha;
	float t_1 = t_0 - 1.0f;
	return t_1 / (logf(powf(t_0, ((float) M_PI))) * (1.0f + ((t_1 * cosTheta) * cosTheta)));
}

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(Float32(1.0) / Float32(Float32(1.0) / alpha)) * alpha)
	t_1 = Float32(t_0 - Float32(1.0))
	return Float32(t_1 / Float32(log((t_0 ^ Float32(pi))) * Float32(Float32(1.0) + Float32(Float32(t_1 * cosTheta) * cosTheta))))
end

MATLAB

function tmp = code(cosTheta, alpha)
	t_0 = (single(1.0) / (single(1.0) / alpha)) * alpha;
	t_1 = t_0 - single(1.0);
	tmp = t_1 / (log((t_0 ^ single(pi))) * (single(1.0) + ((t_1 * 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. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-plus-rev N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-neg N/A

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

   10. inv-pow N/A

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

   11. lower-/.f32 N/A

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

   12. lower-/.f32 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-plus-rev N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-neg N/A

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

   10. inv-pow N/A

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

   11. lower-/.f32 N/A

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

   12. lower-/.f32 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-plus-rev N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-neg N/A

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

   10. inv-pow N/A

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

   11. lower-/.f32 N/A

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

   12. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (log (pow (* alpha alpha) PI)) (+ 1.0 (* (* t_0 cosTheta) 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 + ((t_0 * cosTheta) * 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(Float32(t_0 * cosTheta) * 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) + ((t_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. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 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. Add Preprocessing

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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))) * fma(Float32(t_0 * 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. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 98.6

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

3. Applied rewrites 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	t_0 = Float32(Float32(alpha * alpha) - Float32(1.0))
	return Float32(t_0 / Float32(Float32(pi) * Float32(Float32(log(alpha) * Float32(2.0)) * fma(Float32(cosTheta * cosTheta), t_0, 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. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. pow2 N/A

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

   6. lower-log.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.5%

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

Alternative 5: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(Float32(2.0) * Float32(pi)) * Float32(log(alpha) * fma(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. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. log-pow-rev N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.7

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

3. Applied rewrites 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

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-log.f32 N/A

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

   7. mul-1-neg N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 97.4

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

6. Applied rewrites 97.4%

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

Alternative 6: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / (((log(alpha) * single(pi)) * (single(1.0) - (cosTheta * cosTheta))) * 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. Taylor expanded in alpha around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 97.4%

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

Alternative 7: 95.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((single(pi) * 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. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 95.1%

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

Alternative 8: 95.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = ((alpha * alpha) - single(1.0)) / ((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. Taylor expanded in cosTheta around 0

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

   1. log-pow-rev N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left({\alpha}^{2}\right)}^{\mathsf{PI}\left(\right)}\right)} \]

   2. pow-pow N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\alpha}^{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   3. log-pow-rev N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot 2} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot 2} \]

   9. lower-log.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot 2} \]

   10. lift-PI.f32 95.0

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

4. Applied rewrites 95.0%

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

Alternative 9: 47.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta, alpha)
	tmp = (single(0.5) / (log(alpha) * single(pi))) * (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift--.f32 N/A

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

   4. difference-of-sqr-1 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. pow2 N/A

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

   10. lower-log.f32 N/A

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

   11. lift-+.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift--.f32 N/A

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

3. Applied rewrites 98.1%

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

4. Taylor expanded in cosTheta around 0

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

   1. lift--.f32 94.7

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

6. Applied rewrites 94.7%

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

7. Taylor expanded in alpha around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{\log \alpha \cdot \mathsf{PI}\left(\right)} \cdot \left(\alpha - 1\right) \]

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 47.1

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

9. Applied rewrites 47.1%

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

Reproduce

herbie shell --seed 2025116 
(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)))))