ab-angle->ABCF A

Percentage Accurate: 79.3% → 79.2%

Time: 7.0s

Alternatives: 16

Speedup: N/A×

• 35.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{0.005555555555555556 \cdot angle\_m}\\ {\left(a \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(t\_0 \cdot t\_0, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (* 0.005555555555555556 angle_m))))
   (+
    (pow (* a (sin (* (* PI angle_m) 0.005555555555555556))) 2.0)
    (pow (* b (sin (fma (* t_0 t_0) PI (/ PI 2.0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = sqrt((0.005555555555555556 * angle_m));
	return pow((a * sin(((((double) M_PI) * angle_m) * 0.005555555555555556))), 2.0) + pow((b * sin(fma((t_0 * t_0), ((double) M_PI), (((double) M_PI) / 2.0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = sqrt(Float64(0.005555555555555556 * angle_m))
	return Float64((Float64(a * sin(Float64(Float64(pi * angle_m) * 0.005555555555555556))) ^ 2.0) + (Float64(b * sin(fma(Float64(t_0 * t_0), pi, Float64(pi / 2.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sqrt[N[(0.005555555555555556 * angle$95$m), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(t$95$0 * t$95$0), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   3. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   5. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)}\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)}\right)}^{2} \]

   8. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   10. cos-fabs-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right|\right)}\right)}^{2} \]

   11. sin-+PI/2-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   12. lower-sin.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   14. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   15. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   16. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   17. fabs-mul N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left|\frac{1}{180} \cdot angle\right| \cdot \left|\mathsf{PI}\left(\right)\right|} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   18. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \left|\color{blue}{e^{\log \mathsf{PI}\left(\right)}}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   19. exp-fabs N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{e^{\log \mathsf{PI}\left(\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   20. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{\mathsf{PI}\left(\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

7. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\left|\color{blue}{\frac{1}{180} \cdot angle}\right|, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   2. lift-fabs.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\left|\frac{1}{180} \cdot angle\right|}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   3. rem-sqrt-square-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\left(\frac{1}{180} \cdot angle\right) \cdot \left(\frac{1}{180} \cdot angle\right)}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   4. sqrt-prod N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   5. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \color{blue}{\sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   9. lift-*.f64 79.2

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{\color{blue}{0.005555555555555556 \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

9. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{0.005555555555555556 \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)}^{2} + \frac{b \cdot b}{{\left(\frac{1}{\cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)}\right)}^{2}} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* (* PI angle_m) 0.005555555555555556))) 2.0)
  (/ (* b b) (pow (/ 1.0 (cos (* (* PI 0.005555555555555556) angle_m))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((((double) M_PI) * angle_m) * 0.005555555555555556))), 2.0) + ((b * b) / pow((1.0 / cos(((((double) M_PI) * 0.005555555555555556) * angle_m))), 2.0));
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin(((Math.PI * angle_m) * 0.005555555555555556))), 2.0) + ((b * b) / Math.pow((1.0 / Math.cos(((Math.PI * 0.005555555555555556) * angle_m))), 2.0));
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin(((math.pi * angle_m) * 0.005555555555555556))), 2.0) + ((b * b) / math.pow((1.0 / math.cos(((math.pi * 0.005555555555555556) * angle_m))), 2.0))

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(pi * angle_m) * 0.005555555555555556))) ^ 2.0) + Float64(Float64(b * b) / (Float64(1.0 / cos(Float64(Float64(pi * 0.005555555555555556) * angle_m))) ^ 2.0)))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((pi * angle_m) * 0.005555555555555556))) ^ 2.0) + ((b * b) / ((1.0 / cos(((pi * 0.005555555555555556) * angle_m))) ^ 2.0));
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] / N[Power[N[(1.0 / N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   3. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   5. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)}\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)}\right)}^{2} \]

   8. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   10. cos-fabs-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right|\right)}\right)}^{2} \]

   11. sin-+PI/2-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   12. lower-sin.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   14. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   15. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   16. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   17. fabs-mul N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left|\frac{1}{180} \cdot angle\right| \cdot \left|\mathsf{PI}\left(\right)\right|} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   18. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \left|\color{blue}{e^{\log \mathsf{PI}\left(\right)}}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   19. exp-fabs N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{e^{\log \mathsf{PI}\left(\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   20. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{\mathsf{PI}\left(\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

7. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\left|\color{blue}{\frac{1}{180} \cdot angle}\right|, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   2. lift-fabs.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\left|\frac{1}{180} \cdot angle\right|}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   3. rem-sqrt-square-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\left(\frac{1}{180} \cdot angle\right) \cdot \left(\frac{1}{180} \cdot angle\right)}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   4. sqrt-prod N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   5. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \color{blue}{\sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   9. lift-*.f64 79.2

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{\color{blue}{0.005555555555555556 \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

9. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{0.005555555555555556 \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

11. Applied rewrites 79.2%

    \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + \color{blue}{\frac{b \cdot b}{{\left(\frac{1}{\cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)}\right)}^{2}}} \]
12. Add Preprocessing

Alternative 3: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \sin \left(\left(-t\_0\right) + 0.5 \cdot \pi\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (sin (+ (- t_0) (* 0.5 PI)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	return pow((a * sin(t_0)), 2.0) + pow((b * sin((-t_0 + (0.5 * ((double) M_PI))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (Math.PI * angle_m) * 0.005555555555555556;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.sin((-t_0 + (0.5 * Math.PI)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (math.pi * angle_m) * 0.005555555555555556
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.sin((-t_0 + (0.5 * math.pi)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * sin(Float64(Float64(-t_0) + Float64(0.5 * pi)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = (pi * angle_m) * 0.005555555555555556;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * sin((-t_0 + (0.5 * pi)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[((-t$95$0) + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   3. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   5. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)}\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)}\right)}^{2} \]

   8. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   10. cos-fabs-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right|\right)}\right)}^{2} \]

   11. sin-+PI/2-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   12. lower-sin.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   14. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   15. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   16. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   17. fabs-mul N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left|\frac{1}{180} \cdot angle\right| \cdot \left|\mathsf{PI}\left(\right)\right|} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   18. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \left|\color{blue}{e^{\log \mathsf{PI}\left(\right)}}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   19. exp-fabs N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{e^{\log \mathsf{PI}\left(\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   20. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{\mathsf{PI}\left(\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

7. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}^{2} \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\left|\frac{1}{180} \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}^{2} \]

   2. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\left|\frac{1}{180} \cdot angle\right|, \color{blue}{\mathsf{PI}\left(\right)}, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   3. lift-fma.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)}\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\frac{1}{180} \cdot angle}\right| \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

   5. lift-fabs.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left|\frac{1}{180} \cdot angle\right|} \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

   6. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}^{2} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right)}^{2} \]

   8. sin-+PI/2 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   9. add-exp-log N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{e^{\log \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

   10. exp-fabs N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{\left|e^{\log \mathsf{PI}\left(\right)}\right|}\right)\right)}^{2} \]

   11. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \left|\color{blue}{\mathsf{PI}\left(\right)}\right|\right)\right)}^{2} \]

   12. fabs-mul N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left|\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right|\right)}\right)}^{2} \]

   13. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right|\right)\right)}^{2} \]

   14. metadata-eval N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right|\right)\right)}^{2} \]

   15. mult-flip N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right|\right)\right)}^{2} \]

   16. cos-fabs-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

9. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left(-\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) + 0.5 \cdot \pi\right)}\right)}^{2} \]
10. Add Preprocessing

Alternative 4: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle\_m, 0.005555555555555556, 0.5 \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* (* PI angle_m) 0.005555555555555556))) 2.0)
  (pow (* b (sin (fma (* PI angle_m) 0.005555555555555556 (* 0.5 PI)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((((double) M_PI) * angle_m) * 0.005555555555555556))), 2.0) + pow((b * sin(fma((((double) M_PI) * angle_m), 0.005555555555555556, (0.5 * ((double) M_PI))))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(pi * angle_m) * 0.005555555555555556))) ^ 2.0) + (Float64(b * sin(fma(Float64(pi * angle_m), 0.005555555555555556, Float64(0.5 * pi)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   3. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   5. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)}\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)}\right)}^{2} \]

   8. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   10. cos-fabs-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right|\right)}\right)}^{2} \]

   11. sin-+PI/2-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   12. lower-sin.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   14. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   15. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   16. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   17. fabs-mul N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left|\frac{1}{180} \cdot angle\right| \cdot \left|\mathsf{PI}\left(\right)\right|} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   18. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \left|\color{blue}{e^{\log \mathsf{PI}\left(\right)}}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   19. exp-fabs N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{e^{\log \mathsf{PI}\left(\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   20. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{\mathsf{PI}\left(\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

7. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\left|\color{blue}{\frac{1}{180} \cdot angle}\right|, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   2. lift-fabs.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\left|\frac{1}{180} \cdot angle\right|}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   3. rem-sqrt-square-rev N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\left(\frac{1}{180} \cdot angle\right) \cdot \left(\frac{1}{180} \cdot angle\right)}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   4. sqrt-prod N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   5. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \color{blue}{\sqrt{\frac{1}{180} \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   9. lift-*.f64 79.2

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{\color{blue}{0.005555555555555556 \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]

9. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{0.005555555555555556 \cdot angle} \cdot \sqrt{0.005555555555555556 \cdot angle}}, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} \]
10. Step-by-step derivation

    1. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}, \color{blue}{\mathsf{PI}\left(\right)}, \frac{\pi}{2}\right)\right)\right)}^{2} \]

    2. lift-fma.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}\right) \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)}\right)}^{2} \]

    3. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\frac{1}{180} \cdot angle}\right)} \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

    4. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\sqrt{\color{blue}{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}\right) \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

    5. lift-sqrt.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\color{blue}{\sqrt{\frac{1}{180} \cdot angle}} \cdot \sqrt{\frac{1}{180} \cdot angle}\right) \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

    6. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \sqrt{\color{blue}{\frac{1}{180} \cdot angle}}\right) \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

    7. lift-sqrt.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\sqrt{\frac{1}{180} \cdot angle} \cdot \color{blue}{\sqrt{\frac{1}{180} \cdot angle}}\right) \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

    8. rem-square-sqrt N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

    9. associate-*r* N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)} + \frac{\pi}{2}\right)\right)}^{2} \]

    10. *-commutative N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}} + \frac{\pi}{2}\right)\right)}^{2} \]

    11. lower-fma.f64 N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{1}{180}, \frac{\pi}{2}\right)\right)}\right)}^{2} \]

    12. *-commutative N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{1}{180}, \frac{\pi}{2}\right)\right)\right)}^{2} \]

    13. lift-*.f64 N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{1}{180}, \frac{\pi}{2}\right)\right)\right)}^{2} \]

    14. lift-PI.f64 79.2

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi} \cdot angle, 0.005555555555555556, \frac{\pi}{2}\right)\right)\right)}^{2} \]

    15. lift-PI.f64 N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)\right)}^{2} \]

    16. lift-/.f64 N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right)\right)}^{2} \]

    17. mult-flip N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right)\right)}^{2} \]

    18. metadata-eval N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}^{2} \]

    19. *-commutative N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}^{2} \]

    20. lower-*.f64 N/A

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, \frac{1}{180}, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}^{2} \]

    21. lift-PI.f64 79.2

        \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0.5 \cdot \color{blue}{\pi}\right)\right)\right)}^{2} \]

11. Applied rewrites 79.2%

    \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0.5 \cdot \pi\right)\right)}\right)}^{2} \]
12. Add Preprocessing

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle\_m, 0.5 \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* (* PI angle_m) 0.005555555555555556))) 2.0)
  (pow (* b (sin (fma (* PI 0.005555555555555556) angle_m (* 0.5 PI)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((((double) M_PI) * angle_m) * 0.005555555555555556))), 2.0) + pow((b * sin(fma((((double) M_PI) * 0.005555555555555556), angle_m, (0.5 * ((double) M_PI))))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(pi * angle_m) * 0.005555555555555556))) ^ 2.0) + (Float64(b * sin(fma(Float64(pi * 0.005555555555555556), angle_m, Float64(0.5 * pi)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   3. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   5. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)}\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   7. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)}\right)}^{2} \]

   8. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2} \]

   9. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)}^{2} \]

   10. cos-fabs-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right|\right)}\right)}^{2} \]

   11. sin-+PI/2-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   12. lower-sin.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\left|\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}^{2} \]

   13. lift-PI.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot angle\right)\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   14. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   15. lift-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot angle\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   16. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   17. fabs-mul N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left|\frac{1}{180} \cdot angle\right| \cdot \left|\mathsf{PI}\left(\right)\right|} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   18. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \left|\color{blue}{e^{\log \mathsf{PI}\left(\right)}}\right| + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   19. exp-fabs N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{e^{\log \mathsf{PI}\left(\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

   20. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{\mathsf{PI}\left(\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{2} \]

7. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\left|0.005555555555555556 \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}^{2} \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\left|\frac{1}{180} \cdot angle\right|, \pi, \frac{\pi}{2}\right)\right)}\right)}^{2} \]

   2. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\left|\frac{1}{180} \cdot angle\right|, \color{blue}{\mathsf{PI}\left(\right)}, \frac{\pi}{2}\right)\right)\right)}^{2} \]

   3. lift-fma.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)}\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\color{blue}{\frac{1}{180} \cdot angle}\right| \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

   5. lift-fabs.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left|\frac{1}{180} \cdot angle\right|} \cdot \mathsf{PI}\left(\right) + \frac{\pi}{2}\right)\right)}^{2} \]

   6. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}^{2} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right)}^{2} \]

   8. sin-+PI/2 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   9. add-exp-log N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{e^{\log \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

   10. exp-fabs N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \color{blue}{\left|e^{\log \mathsf{PI}\left(\right)}\right|}\right)\right)}^{2} \]

   11. add-exp-log N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\frac{1}{180} \cdot angle\right| \cdot \left|\color{blue}{\mathsf{PI}\left(\right)}\right|\right)\right)}^{2} \]

   12. fabs-mul N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left|\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right|\right)}\right)}^{2} \]

   13. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right|\right)\right)}^{2} \]

   14. metadata-eval N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right|\right)\right)}^{2} \]

   15. mult-flip N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left|\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right|\right)\right)}^{2} \]

   16. cos-fabs-rev N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

9. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle, 0.5 \cdot \pi\right)\right)}\right)}^{2} \]
10. Add Preprocessing

Alternative 6: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (Math.PI * angle_m) * 0.005555555555555556;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (math.pi * angle_m) * 0.005555555555555556
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = (pi * angle_m) * 0.005555555555555556;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]
6. Add Preprocessing

Alternative 7: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* (* PI angle_m) 0.005555555555555556))) 2.0)
  (pow (* b 1.0) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((((double) M_PI) * angle_m) * 0.005555555555555556))), 2.0) + pow((b * 1.0), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin(((Math.PI * angle_m) * 0.005555555555555556))), 2.0) + Math.pow((b * 1.0), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin(((math.pi * angle_m) * 0.005555555555555556))), 2.0) + math.pow((b * 1.0), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(pi * angle_m) * 0.005555555555555556))) ^ 2.0) + (Float64(b * 1.0) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((pi * angle_m) * 0.005555555555555556))) ^ 2.0) + ((b * 1.0) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

3. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
4. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

   3. lift-/.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. mult-flip N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. metadata-eval N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

   12. lift-PI.f64 79.2

       \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

5. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]

6. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
7. Step-by-step derivation

8. Applied rewrites 79.2%

   \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
9. Add Preprocessing

Alternative 8: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ \mathbf{if}\;a \leq 6 \cdot 10^{-59}:\\ \;\;\;\;\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right), b \cdot b, \left(\left(t\_0 \cdot a\right) \cdot t\_0\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556)))
   (if (<= a 6e-59)
     (*
      (+ 0.5 (* 0.5 (cos (* 2.0 (* PI (* 0.005555555555555556 angle_m))))))
      (* b b))
     (fma (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))) (* b b) (* (* (* t_0 a) t_0) a)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double tmp;
	if (a <= 6e-59) {
		tmp = (0.5 + (0.5 * cos((2.0 * (((double) M_PI) * (0.005555555555555556 * angle_m)))))) * (b * b);
	} else {
		tmp = fma((0.5 + (0.5 * cos((2.0 * t_0)))), (b * b), (((t_0 * a) * t_0) * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	tmp = 0.0
	if (a <= 6e-59)
		tmp = Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle_m)))))) * Float64(b * b));
	else
		tmp = fma(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))), Float64(b * b), Float64(Float64(Float64(t_0 * a) * t_0) * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[a, 6e-59], N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(N[(t$95$0 * a), $MachinePrecision] * t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000002e-59

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]

   if 6.0000000000000002e-59 < a

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. lift-PI.f64 74.5

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. mult-flip N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. metadata-eval N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. *-commutative N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      13. lower-*.f64 74.5

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lower-*.f64 74.5

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b \cdot b, \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_1 := t\_0 \cdot a\\ \mathbf{if}\;a \leq 6 \cdot 10^{-59}:\\ \;\;\;\;\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556)) (t_1 (* t_0 a)))
   (if (<= a 6e-59)
     (*
      (+ 0.5 (* 0.5 (cos (* 2.0 (* PI (* 0.005555555555555556 angle_m))))))
      (* b b))
     (fma t_1 t_1 (* (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))) (* b b))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_1 = t_0 * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = (0.5 + (0.5 * cos((2.0 * (((double) M_PI) * (0.005555555555555556 * angle_m)))))) * (b * b);
	} else {
		tmp = fma(t_1, t_1, ((0.5 + (0.5 * cos((2.0 * t_0)))) * (b * b)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_1 = Float64(t_0 * a)
	tmp = 0.0
	if (a <= 6e-59)
		tmp = Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle_m)))))) * Float64(b * b));
	else
		tmp = fma(t_1, t_1, Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) * Float64(b * b)));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * a), $MachinePrecision]}, If[LessEqual[a, 6e-59], N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$1 + N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000002e-59

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]

   if 6.0000000000000002e-59 < a

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. lift-PI.f64 74.5

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. mult-flip N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. metadata-eval N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. *-commutative N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      13. lower-*.f64 74.5

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lower-*.f64 74.5

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   8. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a, \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\\ t_1 := t\_0 \cdot a\\ \mathbf{if}\;a \leq 6 \cdot 10^{-59}:\\ \;\;\;\;\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI 0.005555555555555556) angle_m)) (t_1 (* t_0 a)))
   (if (<= a 6e-59)
     (*
      (+ 0.5 (* 0.5 (cos (* 2.0 (* PI (* 0.005555555555555556 angle_m))))))
      (* b b))
     (fma t_1 t_1 (* (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))) (* b b))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * 0.005555555555555556) * angle_m;
	double t_1 = t_0 * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = (0.5 + (0.5 * cos((2.0 * (((double) M_PI) * (0.005555555555555556 * angle_m)))))) * (b * b);
	} else {
		tmp = fma(t_1, t_1, ((0.5 + (0.5 * cos((2.0 * t_0)))) * (b * b)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * 0.005555555555555556) * angle_m)
	t_1 = Float64(t_0 * a)
	tmp = 0.0
	if (a <= 6e-59)
		tmp = Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle_m)))))) * Float64(b * b));
	else
		tmp = fma(t_1, t_1, Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) * Float64(b * b)));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * a), $MachinePrecision]}, If[LessEqual[a, 6e-59], N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$1 + N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000002e-59

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]

   if 6.0000000000000002e-59 < a

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. lift-PI.f64 74.5

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. mult-flip N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. metadata-eval N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. *-commutative N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      13. lower-*.f64 74.5

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)\right) \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right) \cdot a\\ \mathbf{if}\;a \leq 6 \cdot 10^{-59}:\\ \;\;\;\;\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0 + {\left(b \cdot 1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* (* PI 0.005555555555555556) angle_m) a)))
   (if (<= a 6e-59)
     (*
      (+ 0.5 (* 0.5 (cos (* 2.0 (* PI (* 0.005555555555555556 angle_m))))))
      (* b b))
     (+ (* t_0 t_0) (pow (* b 1.0) 2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((((double) M_PI) * 0.005555555555555556) * angle_m) * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = (0.5 + (0.5 * cos((2.0 * (((double) M_PI) * (0.005555555555555556 * angle_m)))))) * (b * b);
	} else {
		tmp = (t_0 * t_0) + pow((b * 1.0), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = ((Math.PI * 0.005555555555555556) * angle_m) * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = (0.5 + (0.5 * Math.cos((2.0 * (Math.PI * (0.005555555555555556 * angle_m)))))) * (b * b);
	} else {
		tmp = (t_0 * t_0) + Math.pow((b * 1.0), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = ((math.pi * 0.005555555555555556) * angle_m) * a
	tmp = 0
	if a <= 6e-59:
		tmp = (0.5 + (0.5 * math.cos((2.0 * (math.pi * (0.005555555555555556 * angle_m)))))) * (b * b)
	else:
		tmp = (t_0 * t_0) + math.pow((b * 1.0), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(Float64(pi * 0.005555555555555556) * angle_m) * a)
	tmp = 0.0
	if (a <= 6e-59)
		tmp = Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle_m)))))) * Float64(b * b));
	else
		tmp = Float64(Float64(t_0 * t_0) + (Float64(b * 1.0) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = ((pi * 0.005555555555555556) * angle_m) * a;
	tmp = 0.0;
	if (a <= 6e-59)
		tmp = (0.5 + (0.5 * cos((2.0 * (pi * (0.005555555555555556 * angle_m)))))) * (b * b);
	else
		tmp = (t_0 * t_0) + ((b * 1.0) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, 6e-59], N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000002e-59

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]

   if 6.0000000000000002e-59 < a

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. lift-PI.f64 74.5

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. mult-flip N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. metadata-eval N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. *-commutative N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      13. lower-*.f64 74.5

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lower-*.f64 74.5

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right) \cdot a\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.3%

      \[\leadsto \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 66.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\\ t_1 := t\_0 \cdot a\\ \mathbf{if}\;a \leq 6 \cdot 10^{-59}:\\ \;\;\;\;\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_1 + {\left(b \cdot 1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI 0.005555555555555556) angle_m)) (t_1 (* t_0 a)))
   (if (<= a 6e-59)
     (* (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))) (* b b))
     (+ (* t_1 t_1) (pow (* b 1.0) 2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * 0.005555555555555556) * angle_m;
	double t_1 = t_0 * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = (0.5 + (0.5 * cos((2.0 * t_0)))) * (b * b);
	} else {
		tmp = (t_1 * t_1) + pow((b * 1.0), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (Math.PI * 0.005555555555555556) * angle_m;
	double t_1 = t_0 * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = (0.5 + (0.5 * Math.cos((2.0 * t_0)))) * (b * b);
	} else {
		tmp = (t_1 * t_1) + Math.pow((b * 1.0), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (math.pi * 0.005555555555555556) * angle_m
	t_1 = t_0 * a
	tmp = 0
	if a <= 6e-59:
		tmp = (0.5 + (0.5 * math.cos((2.0 * t_0)))) * (b * b)
	else:
		tmp = (t_1 * t_1) + math.pow((b * 1.0), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * 0.005555555555555556) * angle_m)
	t_1 = Float64(t_0 * a)
	tmp = 0.0
	if (a <= 6e-59)
		tmp = Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) * Float64(b * b));
	else
		tmp = Float64(Float64(t_1 * t_1) + (Float64(b * 1.0) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = (pi * 0.005555555555555556) * angle_m;
	t_1 = t_0 * a;
	tmp = 0.0;
	if (a <= 6e-59)
		tmp = (0.5 + (0.5 * cos((2.0 * t_0)))) * (b * b);
	else
		tmp = (t_1 * t_1) + ((b * 1.0) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * a), $MachinePrecision]}, If[LessEqual[a, 6e-59], N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000002e-59

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. metadata-eval N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. lift-PI.f64 79.2

          \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. Applied rewrites 79.2%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

      3. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. metadata-eval N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

      12. lift-PI.f64 79.2

          \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

   5. Applied rewrites 79.2%

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \color{blue}{{b}^{2}} \]

   8. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)\right) \cdot \left(b \cdot b\right)} \]

   if 6.0000000000000002e-59 < a

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. lift-PI.f64 74.5

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. mult-flip N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. metadata-eval N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. *-commutative N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      13. lower-*.f64 74.5

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lower-*.f64 74.5

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right) \cdot a\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.3%

      \[\leadsto \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 66.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right) \cdot a\\ \mathbf{if}\;a \leq 6 \cdot 10^{-59}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0 + {\left(b \cdot 1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* (* PI 0.005555555555555556) angle_m) a)))
   (if (<= a 6e-59) (* b b) (+ (* t_0 t_0) (pow (* b 1.0) 2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((((double) M_PI) * 0.005555555555555556) * angle_m) * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = b * b;
	} else {
		tmp = (t_0 * t_0) + pow((b * 1.0), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = ((Math.PI * 0.005555555555555556) * angle_m) * a;
	double tmp;
	if (a <= 6e-59) {
		tmp = b * b;
	} else {
		tmp = (t_0 * t_0) + Math.pow((b * 1.0), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = ((math.pi * 0.005555555555555556) * angle_m) * a
	tmp = 0
	if a <= 6e-59:
		tmp = b * b
	else:
		tmp = (t_0 * t_0) + math.pow((b * 1.0), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(Float64(pi * 0.005555555555555556) * angle_m) * a)
	tmp = 0.0
	if (a <= 6e-59)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(t_0 * t_0) + (Float64(b * 1.0) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = ((pi * 0.005555555555555556) * angle_m) * a;
	tmp = 0.0;
	if (a <= 6e-59)
		tmp = b * b;
	else
		tmp = (t_0 * t_0) + ((b * 1.0) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, 6e-59], N[(b * b), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000002e-59

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      2. lower-*.f64 57.1

         \[\leadsto b \cdot \color{blue}{b} \]

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 6.0000000000000002e-59 < a

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. metadata-eval N/A

         \[\leadsto {\left(a \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. lift-PI.f64 74.5

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. mult-flip N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. metadata-eval N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. *-commutative N/A

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      13. lower-*.f64 74.5

          \[\leadsto {\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lower-*.f64 74.5

         \[\leadsto \color{blue}{\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot \frac{1}{180}\right) \cdot angle\right) \cdot a\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.3%

      \[\leadsto \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 66.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-59}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, {\left(angle\_m \cdot \left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 6e-59)
   (* b b)
   (fma b b (pow (* angle_m (* (* PI a) 0.005555555555555556)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 6e-59) {
		tmp = b * b;
	} else {
		tmp = fma(b, b, pow((angle_m * ((((double) M_PI) * a) * 0.005555555555555556)), 2.0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 6e-59)
		tmp = Float64(b * b);
	else
		tmp = fma(b, b, (Float64(angle_m * Float64(Float64(pi * a) * 0.005555555555555556)) ^ 2.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 6e-59], N[(b * b), $MachinePrecision], N[(b * b + N[Power[N[(angle$95$m * N[(N[(Pi * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000002e-59

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      2. lower-*.f64 57.1

         \[\leadsto b \cdot \color{blue}{b} \]

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 6.0000000000000002e-59 < a

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. metadata-eval N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. lift-PI.f64 79.2

          \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   3. Applied rewrites 79.2%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} \]

      3. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. mult-flip N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. metadata-eval N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)}\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right)\right)}^{2} \]

      12. lift-PI.f64 79.2

          \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

   5. Applied rewrites 79.2%

      \[\leadsto {\left(a \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}^{2} \]

   7. Applied rewrites 45.1%

      \[\leadsto \color{blue}{{\left(e^{2}\right)}^{\log \left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)}} + {\left(b \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + {b}^{2}} \]

   10. Applied rewrites 74.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, {\left(angle \cdot \left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right)\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ \mathbf{if}\;{\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)))
   (if (<= (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0)) 5e+303)
     (* b b)
     (sqrt (* (* b b) (* b b))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double tmp;
	if ((pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0)) <= 5e+303) {
		tmp = b * b;
	} else {
		tmp = sqrt(((b * b) * (b * b)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double tmp;
	if ((Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0)) <= 5e+303) {
		tmp = b * b;
	} else {
		tmp = Math.sqrt(((b * b) * (b * b)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (angle_m / 180.0) * math.pi
	tmp = 0
	if (math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)) <= 5e+303:
		tmp = b * b
	else:
		tmp = math.sqrt(((b * b) * (b * b)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	tmp = 0.0
	if (Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0)) <= 5e+303)
		tmp = Float64(b * b);
	else
		tmp = sqrt(Float64(Float64(b * b) * Float64(b * b)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = (angle_m / 180.0) * pi;
	tmp = 0.0;
	if ((((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0)) <= 5e+303)
		tmp = b * b;
	else
		tmp = sqrt(((b * b) * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 5e+303], N[(b * b), $MachinePrecision], N[Sqrt[N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) < 4.9999999999999997e303

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      2. lower-*.f64 57.1

         \[\leadsto b \cdot \color{blue}{b} \]

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 4.9999999999999997e303 < (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)))

   1. Initial program 79.3%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      2. lower-*.f64 57.1

         \[\leadsto b \cdot \color{blue}{b} \]

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{b \cdot b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{b} \]

      2. pow2 N/A

         \[\leadsto {b}^{\color{blue}{2}} \]

      3. pow-to-exp N/A

         \[\leadsto e^{\log b \cdot 2} \]

      4. lower-exp.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      6. lower-log.f64 27.1

         \[\leadsto e^{\log b \cdot 2} \]

   6. Applied rewrites 27.1%

      \[\leadsto e^{\log b \cdot 2} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      4. exp-fabs N/A

         \[\leadsto \left|e^{\log b \cdot 2}\right| \]

      5. pow-to-exp N/A

         \[\leadsto \left|{b}^{2}\right| \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{{b}^{2} \cdot {b}^{2}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{{b}^{2} \cdot {b}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{{b}^{2} \cdot {b}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\left(b \cdot b\right) \cdot {b}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(b \cdot b\right) \cdot {b}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]

      12. lower-*.f64 49.5

          \[\leadsto \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]

   8. Applied rewrites 49.5%

      \[\leadsto \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.1% accurate, 29.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* b b))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return b * b;
}

Fortran

angle_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = b * b
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return b * b;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return b * b

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(b * b)
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = b * b;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{b}^{2}} \]
3. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto b \cdot \color{blue}{b} \]

   2. lower-*.f64 57.1

      \[\leadsto b \cdot \color{blue}{b} \]

4. Applied rewrites 57.1%

   \[\leadsto \color{blue}{b \cdot b} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))