ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.6%

Time: 5.4s

Alternatives: 12

Speedup: 1.0×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi}\\ {\left(a \cdot \cos \left(t\_0 \cdot t\_0\right)\right)}^{2} + {\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot b}\right)}^{-2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (* (* 0.005555555555555556 angle_m) PI))))
   (+
    (pow (* a (cos (* t_0 t_0))) 2.0)
    (pow (/ -1.0 (* (sin (* -0.005555555555555556 (* angle_m PI))) b)) -2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = sqrt(((0.005555555555555556 * angle_m) * ((double) M_PI)));
	return pow((a * cos((t_0 * t_0))), 2.0) + pow((-1.0 / (sin((-0.005555555555555556 * (angle_m * ((double) M_PI)))) * b)), -2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.sqrt(((0.005555555555555556 * angle_m) * Math.PI));
	return Math.pow((a * Math.cos((t_0 * t_0))), 2.0) + Math.pow((-1.0 / (Math.sin((-0.005555555555555556 * (angle_m * Math.PI))) * b)), -2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.sqrt(((0.005555555555555556 * angle_m) * math.pi))
	return math.pow((a * math.cos((t_0 * t_0))), 2.0) + math.pow((-1.0 / (math.sin((-0.005555555555555556 * (angle_m * math.pi))) * b)), -2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = sqrt(Float64(Float64(0.005555555555555556 * angle_m) * pi))
	return Float64((Float64(a * cos(Float64(t_0 * t_0))) ^ 2.0) + (Float64(-1.0 / Float64(sin(Float64(-0.005555555555555556 * Float64(angle_m * pi))) * b)) ^ -2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = sqrt(((0.005555555555555556 * angle_m) * pi));
	tmp = ((a * cos((t_0 * t_0))) ^ 2.0) + ((-1.0 / (sin((-0.005555555555555556 * (angle_m * pi))) * b)) ^ -2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(-1.0 / N[(N[Sin[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.6%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sub N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. mult-flip N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-unsound-pow.f64 78.3

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{\color{blue}{{\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{-1}}} \]

3. Applied rewrites 79.6%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{-1}}} \]

5. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. div-flip-rev N/A

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

   7. lift-/.f64 N/A

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

   8. inv-pow N/A

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

   9. sqr-pow N/A

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

   10. lower-unsound-*.f64 N/A

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

   11. lower-unsound-pow.f64 N/A

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

   12. lower-unsound-/.f64 N/A

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

   13. lower-unsound-pow.f64 N/A

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

   14. lower-unsound-/.f64 79.5

       \[\leadsto {\left(a \cdot \cos \left({\left(\frac{180}{angle \cdot \pi}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{180}{angle \cdot \pi}\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)}^{2} + {\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2} \]

7. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\frac{180}{angle \cdot \pi}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{180}{angle \cdot \pi}\right)}^{\left(\frac{-1}{2}\right)}\right)}\right)}^{2} + {\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. sqr-abs-rev N/A

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

   3. lower-*.f64 N/A

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

9. Applied rewrites 79.6%

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

Alternative 2: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} + {\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot b}\right)}^{-2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle_m 180.0)))) 2.0)
  (pow (/ -1.0 (* (sin (* -0.005555555555555556 (* angle_m PI))) b)) -2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((((double) M_PI) * (angle_m / 180.0)))), 2.0) + pow((-1.0 / (sin((-0.005555555555555556 * (angle_m * ((double) M_PI)))) * b)), -2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos((Math.PI * (angle_m / 180.0)))), 2.0) + Math.pow((-1.0 / (Math.sin((-0.005555555555555556 * (angle_m * Math.PI))) * b)), -2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos((math.pi * (angle_m / 180.0)))), 2.0) + math.pow((-1.0 / (math.sin((-0.005555555555555556 * (angle_m * math.pi))) * b)), -2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0) + (Float64(-1.0 / Float64(sin(Float64(-0.005555555555555556 * Float64(angle_m * pi))) * b)) ^ -2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * cos((pi * (angle_m / 180.0)))) ^ 2.0) + ((-1.0 / (sin((-0.005555555555555556 * (angle_m * pi))) * b)) ^ -2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(-1.0 / N[(N[Sin[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sub N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. mult-flip N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-unsound-pow.f64 78.3

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{\color{blue}{{\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{-1}}} \]

3. Applied rewrites 79.6%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{-1}}} \]

5. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2}} \]
6. Add Preprocessing

Alternative 3: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. mult-flip N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.5

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

3. Applied rewrites 79.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. mult-flip N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.6

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

5. Applied rewrites 79.6%

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

Alternative 4: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 79.6

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

3. Applied rewrites 79.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 79.6

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

5. Applied rewrites 79.6%

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

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right), a \cdot a, {\left(\frac{-1}{\sin \left(\left(-0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot b}\right)}^{-2}\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (+ 0.5 (* 0.5 (cos (* 2.0 (* (* 0.005555555555555556 angle_m) PI)))))
  (* a a)
  (pow (/ -1.0 (* (sin (* (* -0.005555555555555556 angle_m) PI)) b)) -2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return fma((0.5 + (0.5 * cos((2.0 * ((0.005555555555555556 * angle_m) * ((double) M_PI)))))), (a * a), pow((-1.0 / (sin(((-0.005555555555555556 * angle_m) * ((double) M_PI))) * b)), -2.0));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return fma(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle_m) * pi))))), Float64(a * a), (Float64(-1.0 / Float64(sin(Float64(Float64(-0.005555555555555556 * angle_m) * pi)) * b)) ^ -2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[Power[N[(-1.0 / N[(N[Sin[N[(N[(-0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sub N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. mult-flip N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-unsound-pow.f64 78.3

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{\color{blue}{{\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{-1}}} \]

3. Applied rewrites 79.6%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{-1}}} \]

5. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. div-flip-rev N/A

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

   7. lift-/.f64 N/A

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

   8. inv-pow N/A

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

   9. sqr-pow N/A

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

   10. lower-unsound-*.f64 N/A

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

   11. lower-unsound-pow.f64 N/A

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

   12. lower-unsound-/.f64 N/A

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

   13. lower-unsound-pow.f64 N/A

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

   14. lower-unsound-/.f64 79.5

       \[\leadsto {\left(a \cdot \cos \left({\left(\frac{180}{angle \cdot \pi}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{180}{angle \cdot \pi}\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)\right)}^{2} + {\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2} \]

7. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\frac{180}{angle \cdot \pi}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{180}{angle \cdot \pi}\right)}^{\left(\frac{-1}{2}\right)}\right)}\right)}^{2} + {\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2} \]

9. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right), a \cdot a, {\left(\frac{-1}{\sin \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b}\right)}^{-2}\right)} \]
10. Add Preprocessing

Alternative 6: 79.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sub N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. mult-flip N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-unsound-pow.f64 78.3

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{\color{blue}{{\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{-1}}} \]

3. Applied rewrites 79.6%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{1}}{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{-1}}} \]

5. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{{\left(\frac{-1}{\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b}\right)}^{-2}} \]

6. Taylor expanded in angle around 0

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

8. Applied rewrites 79.6%

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

Alternative 7: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a 1.0) 2.0) (pow (* b (sin (* PI (/ angle_m 180.0)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * 1.0), 2.0) + pow((b * sin((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * 1.0), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle_m / 180.0)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * 1.0), 2.0) + math.pow((b * math.sin((math.pi * (angle_m / 180.0)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * 1.0) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * 1.0) ^ 2.0) + ((b * sin((pi * (angle_m / 180.0)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 79.6%

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

Alternative 8: 65.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(angle\_m \cdot angle\_m\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \pi\right) \cdot \pi\right) \cdot b, b, \left(\mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right), 0.5, 0.5\right) \cdot a\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.8e-54)
   (* a a)
   (fma
    (* (* (* (* (* angle_m angle_m) 3.08641975308642e-5) PI) PI) b)
    b
    (* (* (fma (cos (* -0.011111111111111112 (* PI angle_m))) 0.5 0.5) a) a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.8e-54) {
		tmp = a * a;
	} else {
		tmp = fma((((((angle_m * angle_m) * 3.08641975308642e-5) * ((double) M_PI)) * ((double) M_PI)) * b), b, ((fma(cos((-0.011111111111111112 * (((double) M_PI) * angle_m))), 0.5, 0.5) * a) * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.8e-54)
		tmp = Float64(a * a);
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(angle_m * angle_m) * 3.08641975308642e-5) * pi) * pi) * b), b, Float64(Float64(fma(cos(Float64(-0.011111111111111112 * Float64(pi * angle_m))), 0.5, 0.5) * a) * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.8e-54], N[(a * a), $MachinePrecision], N[(N[(N[(N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(N[Cos[N[(-0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.79999999999999988e-54

   1. Initial program 79.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 1.79999999999999988e-54 < b

   1. Initial program 79.6%

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

   3. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), b \cdot b, \left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b \cdot b, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, b \cdot b, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), b \cdot b, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right), b \cdot b, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right), b \cdot b, \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a\right) \]

      5. lower-PI.f64 64.2

         \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right), b \cdot b, \left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a\right) \]

   6. Applied rewrites 64.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)}, b \cdot b, \left(\left(0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a\right) \]

   8. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \pi\right) \cdot \pi\right) \cdot b, b, \left(\mathsf{fma}\left(\cos \left(-0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot a\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot angle\_m, angle\_m, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 5.2e+72)
   (fma
    (*
     (*
      (* PI PI)
      (fma -3.08641975308642e-5 (* a a) (* (* b b) 3.08641975308642e-5)))
     angle_m)
    angle_m
    (* a a))
   (* a a)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 5.2e+72) {
		tmp = fma((((((double) M_PI) * ((double) M_PI)) * fma(-3.08641975308642e-5, (a * a), ((b * b) * 3.08641975308642e-5))) * angle_m), angle_m, (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 5.2e+72)
		tmp = fma(Float64(Float64(Float64(pi * pi) * fma(-3.08641975308642e-5, Float64(a * a), Float64(Float64(b * b) * 3.08641975308642e-5))) * angle_m), angle_m, Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 5.2e+72], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(-3.08641975308642e-5 * N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * angle$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.19999999999999963e72

   1. Initial program 79.6%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({angle}^{2}, \color{blue}{\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, {a}^{2}\right) \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, {a}^{2} \cdot {\pi}^{2}, 3.08641975308642 \cdot 10^{-5} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right), {a}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto {angle}^{2} \cdot \mathsf{fma}\left(\frac{-1}{32400}, {a}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right) + \color{blue}{{a}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {a}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{a}}^{2} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {a}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right) \cdot {angle}^{2} + {a}^{2} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {a}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(angle \cdot angle\right) + {a}^{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400}, {a}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right) \cdot angle\right) \cdot angle + {\color{blue}{a}}^{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{32400}, {a}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right) \cdot angle, \color{blue}{angle}, {a}^{2}\right) \]

   6. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot angle, \color{blue}{angle}, a \cdot a\right) \]

   if 5.19999999999999963e72 < a

   1. Initial program 79.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3e+72)
   (fma
    (* angle_m angle_m)
    (*
     (* PI PI)
     (fma -3.08641975308642e-5 (* a a) (* (* b b) 3.08641975308642e-5)))
    (* a a))
   (* a a)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3e+72) {
		tmp = fma((angle_m * angle_m), ((((double) M_PI) * ((double) M_PI)) * fma(-3.08641975308642e-5, (a * a), ((b * b) * 3.08641975308642e-5))), (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3e+72)
		tmp = fma(Float64(angle_m * angle_m), Float64(Float64(pi * pi) * fma(-3.08641975308642e-5, Float64(a * a), Float64(Float64(b * b) * 3.08641975308642e-5))), Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3e+72], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(-3.08641975308642e-5 * N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.00000000000000003e72

   1. Initial program 79.6%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({angle}^{2}, \color{blue}{\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, {a}^{2}\right) \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, {a}^{2} \cdot {\pi}^{2}, 3.08641975308642 \cdot 10^{-5} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right), {a}^{2}\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 41.4%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \color{blue}{\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}, a \cdot a\right) \]

   if 3.00000000000000003e72 < a

   1. Initial program 79.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.3% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

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 = a * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 57.1

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

4. Applied rewrites 57.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 57.1

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

6. Applied rewrites 57.1%

   \[\leadsto \color{blue}{a \cdot a} \]
7. Add Preprocessing

Alternative 12: 53.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ \mathbf{if}\;{\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \leq \infty:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{{a}^{8}}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (if (<= (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)) INFINITY)
     (* a a)
     (sqrt (sqrt (pow a 8.0))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0)) <= ((double) INFINITY)) {
		tmp = a * a;
	} else {
		tmp = sqrt(sqrt(pow(a, 8.0)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double tmp;
	if ((Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0)) <= Double.POSITIVE_INFINITY) {
		tmp = a * a;
	} else {
		tmp = Math.sqrt(Math.sqrt(Math.pow(a, 8.0)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (angle_m / 180.0)
	tmp = 0
	if (math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)) <= math.inf:
		tmp = a * a
	else:
		tmp = math.sqrt(math.sqrt(math.pow(a, 8.0)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0)) <= Inf)
		tmp = Float64(a * a);
	else
		tmp = sqrt(sqrt((a ^ 8.0)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	tmp = 0.0;
	if ((((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0)) <= Inf)
		tmp = a * a;
	else
		tmp = sqrt(sqrt((a ^ 8.0)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], Infinity], N[(a * a), $MachinePrecision], N[Sqrt[N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < +inf.0

   1. Initial program 79.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if +inf.0 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 79.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

      \[\leadsto \color{blue}{a \cdot a} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{a \cdot a} \cdot \color{blue}{\sqrt{a \cdot a}} \]

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 49.8

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

   8. Applied rewrites 49.8%

      \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-prod-up N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(\left(2 + 2\right) + \left(2 + 2\right)\right)}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(\left(2 + 2\right) + \left(2 + 2\right)\right)}}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(4 + \left(2 + 2\right)\right)}}} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(4 + 4\right)}}} \]

      15. metadata-eval 45.0

          \[\leadsto \sqrt{\sqrt{{a}^{8}}} \]

   10. Applied rewrites 45.0%

       \[\leadsto \sqrt{\sqrt{{a}^{8}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))