GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 3.9s

Alternatives: 12

Speedup: 0.8×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (*
     (log (pow (* (* alpha alpha) (* alpha alpha)) (/ PI 2.0)))
     (+ 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) * (alpha * alpha)), (((float) M_PI) / 2.0f))) * (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(Float32(alpha * alpha) * Float32(alpha * alpha)) ^ Float32(Float32(pi) / Float32(2.0)))) * 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) * (alpha * alpha)) ^ (single(pi) / single(2.0)))) * (single(1.0) + ((t_0 * cosTheta) * cosTheta)));
end

Derivation

1. Initial program 98.5%

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

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

   2. pow2 N/A

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

   3. lift-PI.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. sqr-pow N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. lift-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lift-PI.f32 98.7

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

5. Applied rewrites 98.7%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right)}^{\left(\frac{\pi}{2}\right)}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
6. 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 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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: 98.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in cosTheta around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 48.9%

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

5. Taylor expanded in cosTheta around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

7. Applied rewrites 98.5%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/ (/ t_0 (* (* (log alpha) PI) 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 / ((logf(alpha) * ((float) M_PI)) * 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(Float32(t_0 / Float32(Float32(log(alpha) * Float32(pi)) * Float32(2.0))) / fma(Float32(cosTheta * cosTheta), t_0, Float32(1.0)))
end

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.4%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 98.5%

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

Alternative 6: 98.4% 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 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 7: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f32 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 97.4

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

4. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lift-log.f32 N/A

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

   11. +-commutative N/A

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

   12. pow2 N/A

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

   13. mul-1-neg N/A

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

   14. lower-*.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. mul-1-neg N/A

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

   17. pow2 N/A

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

   18. +-commutative N/A

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

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

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

   20. lower-fma.f32 N/A

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

   21. lower-neg.f32 97.4

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

6. Applied rewrites 97.4%

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

Alternative 8: 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 98.5%

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

2. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f32 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 97.4

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

4. Applied rewrites 97.4%

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

5. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower--.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 97.4

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

7. Applied rewrites 97.4%

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

Alternative 9: 95.3% 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 98.5%

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

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 95.3%

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

Alternative 10: 95.2% 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 98.5%

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

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 95.2

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

4. Applied rewrites 95.2%

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

Alternative 11: 1.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf

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

   1. lower-/.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. log-rec N/A

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

   9. lower-neg.f32 N/A

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

   10. lower-log.f32 1.9

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

4. Applied rewrites 1.9%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. frac-2neg N/A

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

   13. lower-/.f32 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

6. Applied rewrites 1.9%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 1.9

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

8. Applied rewrites 1.9%

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

Alternative 12: 1.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in alpha around inf

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

   1. lower-/.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. log-rec N/A

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

   9. lower-neg.f32 N/A

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

   10. lower-log.f32 1.9

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

4. Applied rewrites 1.9%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. frac-2neg N/A

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

   13. lower-/.f32 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

6. Applied rewrites 1.9%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 1.9

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

8. Applied rewrites 1.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-log.f32 1.9

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

10. Applied rewrites 1.9%

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

Reproduce

herbie shell --seed 2025101 
(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)))))