ab-angle->ABCF C

Percentage Accurate: 80.0% → 79.9%

Time: 1.2min

Alternatives: 15

Speedup: 1.0×

• 37.1% 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: 80.0% 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: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)}\\ t_1 := angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\\ {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{t\_1}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log t\_1}\right) \cdot \left({\left({t\_0}^{2}\right)}^{0.16666666666666666} \cdot {t\_0}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cbrt (* PI (* angle_m 0.005555555555555556))))
        (t_1 (* angle_m (* 0.005555555555555556 PI))))
   (+
    (pow
     (*
      a
      (cos
       (*
        (* (pow (cbrt t_1) 2.0) (exp (* 0.16666666666666666 (log t_1))))
        (*
         (pow (pow t_0 2.0) 0.16666666666666666)
         (pow t_0 0.16666666666666666)))))
     2.0)
    (pow (* b (sin (/ PI (/ 180.0 angle_m)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = cbrt((((double) M_PI) * (angle_m * 0.005555555555555556)));
	double t_1 = angle_m * (0.005555555555555556 * ((double) M_PI));
	return pow((a * cos(((pow(cbrt(t_1), 2.0) * exp((0.16666666666666666 * log(t_1)))) * (pow(pow(t_0, 2.0), 0.16666666666666666) * pow(t_0, 0.16666666666666666))))), 2.0) + pow((b * sin((((double) M_PI) / (180.0 / angle_m)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.cbrt((Math.PI * (angle_m * 0.005555555555555556)));
	double t_1 = angle_m * (0.005555555555555556 * Math.PI);
	return Math.pow((a * Math.cos(((Math.pow(Math.cbrt(t_1), 2.0) * Math.exp((0.16666666666666666 * Math.log(t_1)))) * (Math.pow(Math.pow(t_0, 2.0), 0.16666666666666666) * Math.pow(t_0, 0.16666666666666666))))), 2.0) + Math.pow((b * Math.sin((Math.PI / (180.0 / angle_m)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = cbrt(Float64(pi * Float64(angle_m * 0.005555555555555556)))
	t_1 = Float64(angle_m * Float64(0.005555555555555556 * pi))
	return Float64((Float64(a * cos(Float64(Float64((cbrt(t_1) ^ 2.0) * exp(Float64(0.16666666666666666 * log(t_1)))) * Float64(((t_0 ^ 2.0) ^ 0.16666666666666666) * (t_0 ^ 0.16666666666666666))))) ^ 2.0) + (Float64(b * sin(Float64(pi / Float64(180.0 / angle_m)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Power[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(N[(N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision] * N[Power[t$95$0, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. metadata-eval 82.6%

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

   2. div-inv 82.6%

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

   3. clear-num 82.6%

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

   4. un-div-inv 82.7%

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

6. Applied egg-rr 82.7%

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

   1. metadata-eval 82.7%

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

   2. div-inv 82.7%

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

   3. add-cube-cbrt 82.6%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \frac{angle}{180}} \cdot \sqrt[3]{\pi \cdot \frac{angle}{180}}\right) \cdot \sqrt[3]{\pi \cdot \frac{angle}{180}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   4. pow3 82.6%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \frac{angle}{180}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   5. pow-to-exp 37.9%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\sqrt[3]{\pi \cdot \frac{angle}{180}}\right) \cdot 3}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   6. div-inv 37.9%

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\sqrt[3]{\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}}\right) \cdot 3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   7. metadata-eval 37.9%

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\sqrt[3]{\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)}\right) \cdot 3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

8. Applied egg-rr 37.9%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot 3}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 37.9%

    \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot {\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{0.16666666666666666}\right) \cdot {\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
11. Step-by-step derivation

    1. add-exp-log 37.9%

       \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot \color{blue}{e^{\log \left({\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{0.16666666666666666}\right)}}\right) \cdot {\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

    2. log-pow 37.9%

       \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{\color{blue}{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right) \cdot {\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

12. Applied egg-rr 37.9%

    \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot \color{blue}{e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right) \cdot {\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
13. Step-by-step derivation

    1. add-cube-cbrt 37.9%

       \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot {\color{blue}{\left(\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)} \cdot \sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right) \cdot \sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

    2. unpow2 37.9%

       \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot {\left(\color{blue}{{\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2}} \cdot \sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{0.16666666666666666}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

    3. unpow-prod-down 37.9%

       \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \color{blue}{\left({\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{0.16666666666666666}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

14. Applied egg-rr 37.9%

    \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \color{blue}{\left({\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{0.16666666666666666}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
15. Step-by-step derivation

    1. associate-*r* 37.9%

       \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \left({\left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}}\right)}^{2}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

    2. associate-*r* 37.9%

       \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \left({\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{2}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}}\right)}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

16. Simplified 37.9%

    \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \color{blue}{\left({\left({\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{2}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right)}^{0.16666666666666666}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

17. Final simplification 37.9%

    \[\leadsto {\left(a \cdot \cos \left(\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{2} \cdot e^{0.16666666666666666 \cdot \log \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \left({\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2}\right)}^{0.16666666666666666} \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
18. Add Preprocessing

Alternative 2: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)}^{2} + {\left(a \cdot \cos \left(e^{\log \left(\sqrt[3]{\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)}\right) \cdot 3}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ 180.0 angle_m)))) 2.0)
  (pow
   (*
    a
    (cos (exp (* (log (cbrt (* PI (* angle_m 0.005555555555555556)))) 3.0))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((((double) M_PI) / (180.0 / angle_m)))), 2.0) + pow((a * cos(exp((log(cbrt((((double) M_PI) * (angle_m * 0.005555555555555556)))) * 3.0)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((b * Math.sin((Math.PI / (180.0 / angle_m)))), 2.0) + Math.pow((a * Math.cos(Math.exp((Math.log(Math.cbrt((Math.PI * (angle_m * 0.005555555555555556)))) * 3.0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi / Float64(180.0 / angle_m)))) ^ 2.0) + (Float64(a * cos(exp(Float64(log(cbrt(Float64(pi * Float64(angle_m * 0.005555555555555556)))) * 3.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[Exp[N[(N[Log[N[Power[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. metadata-eval 82.6%

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

   2. div-inv 82.6%

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

   3. clear-num 82.6%

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

   4. un-div-inv 82.7%

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

6. Applied egg-rr 82.7%

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

   1. metadata-eval 82.7%

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

   2. div-inv 82.7%

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

   3. add-cube-cbrt 82.6%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \frac{angle}{180}} \cdot \sqrt[3]{\pi \cdot \frac{angle}{180}}\right) \cdot \sqrt[3]{\pi \cdot \frac{angle}{180}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   4. pow3 82.6%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \frac{angle}{180}}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   5. pow-to-exp 37.9%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\sqrt[3]{\pi \cdot \frac{angle}{180}}\right) \cdot 3}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   6. div-inv 37.9%

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\sqrt[3]{\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}}\right) \cdot 3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

   7. metadata-eval 37.9%

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \left(\sqrt[3]{\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)}\right) \cdot 3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

8. Applied egg-rr 37.9%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot 3}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

9. Final simplification 37.9%

   \[\leadsto {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(e^{\log \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot 3}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. metadata-eval 82.6%

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

   2. div-inv 82.6%

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

   3. clear-num 82.6%

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

   4. un-div-inv 82.7%

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

6. Applied egg-rr 82.7%

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

   1. metadata-eval 82.7%

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

   2. div-inv 82.7%

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

   3. clear-num 82.7%

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

8. Applied egg-rr 82.7%

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

9. Final simplification 82.7%

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

Alternative 4: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. metadata-eval 82.6%

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

   2. div-inv 82.6%

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

   3. clear-num 82.6%

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

   4. un-div-inv 82.7%

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

6. Applied egg-rr 82.7%

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

   1. metadata-eval 82.7%

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

   2. div-inv 82.7%

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

   3. clear-num 82.7%

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

   4. div-inv 82.6%

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

   5. associate-/r* 82.7%

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

   6. metadata-eval 82.7%

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

8. Applied egg-rr 82.7%

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

9. Final simplification 82.7%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. metadata-eval 82.6%

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

   2. div-inv 82.6%

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

   3. clear-num 82.6%

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

   4. un-div-inv 82.7%

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

6. Applied egg-rr 82.7%

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

7. Taylor expanded in angle around inf 82.5%

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

   1. associate-*r* 82.7%

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

   2. *-commutative 82.7%

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

   3. associate-*r* 82.7%

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

9. Simplified 82.7%

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

10. Final simplification 82.7%

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

Alternative 6: 79.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. metadata-eval 82.6%

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

   2. div-inv 82.6%

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

   3. clear-num 82.6%

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

   4. un-div-inv 82.7%

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

6. Applied egg-rr 82.7%

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

7. Taylor expanded in angle around 0 82.6%

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

8. Final simplification 82.6%

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

Alternative 7: 79.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Final simplification 82.5%

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

Alternative 8: 79.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Taylor expanded in angle around inf 82.5%

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

7. Final simplification 82.5%

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

Alternative 9: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-234}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle\_m \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.8e-234)
   (+
    (pow a 2.0)
    (*
     angle_m
     (*
      (* 0.005555555555555556 PI)
      (* angle_m (* b (* PI (* 0.005555555555555556 b)))))))
   (+ (pow a 2.0) (pow (* b (* 0.005555555555555556 (* angle_m PI))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.8e-234) {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (angle_m * (b * (((double) M_PI) * (0.005555555555555556 * b))))));
	} else {
		tmp = pow(a, 2.0) + pow((b * (0.005555555555555556 * (angle_m * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.8e-234) {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (angle_m * (b * (Math.PI * (0.005555555555555556 * b))))));
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (0.005555555555555556 * (angle_m * Math.PI))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 2.8e-234:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (angle_m * (b * (math.pi * (0.005555555555555556 * b))))))
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * (0.005555555555555556 * (angle_m * math.pi))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.8e-234)
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(angle_m * Float64(b * Float64(pi * Float64(0.005555555555555556 * b)))))));
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(0.005555555555555556 * Float64(angle_m * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.8e-234)
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (angle_m * (b * (pi * (0.005555555555555556 * b))))));
	else
		tmp = (a ^ 2.0) + ((b * (0.005555555555555556 * (angle_m * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.8e-234], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(angle$95$m * N[(b * N[(Pi * N[(0.005555555555555556 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999999e-234

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in angle around 0 83.2%

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

   6. Taylor expanded in angle around 0 75.5%

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

      1. unpow2 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*l* 76.3%

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

      4. associate-*r* 76.3%

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

      5. *-commutative 76.3%

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

      6. associate-*l* 76.3%

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

      7. associate-*r* 76.3%

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

      8. *-commutative 76.3%

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

      9. associate-*r* 76.3%

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

      10. *-commutative 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. associate-*l* 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

   10. Simplified 75.8%

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

   if 2.7999999999999999e-234 < b

   1. Initial program 81.5%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in angle around 0 81.3%

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

   6. Taylor expanded in angle around 0 77.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-234}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-234}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle\_m \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle\_m \cdot \left(\pi \cdot b\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.8e-234)
   (+
    (pow a 2.0)
    (*
     angle_m
     (*
      (* 0.005555555555555556 PI)
      (* angle_m (* b (* PI (* 0.005555555555555556 b)))))))
   (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* angle_m (* PI b)) 2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.8e-234) {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (angle_m * (b * (((double) M_PI) * (0.005555555555555556 * b))))));
	} else {
		tmp = pow(a, 2.0) + (3.08641975308642e-5 * pow((angle_m * (((double) M_PI) * b)), 2.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.8e-234) {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (angle_m * (b * (Math.PI * (0.005555555555555556 * b))))));
	} else {
		tmp = Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((angle_m * (Math.PI * b)), 2.0));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 2.8e-234:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (angle_m * (b * (math.pi * (0.005555555555555556 * b))))))
	else:
		tmp = math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((angle_m * (math.pi * b)), 2.0))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.8e-234)
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(angle_m * Float64(b * Float64(pi * Float64(0.005555555555555556 * b)))))));
	else
		tmp = Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(angle_m * Float64(pi * b)) ^ 2.0)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.8e-234)
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (angle_m * (b * (pi * (0.005555555555555556 * b))))));
	else
		tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((angle_m * (pi * b)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.8e-234], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(angle$95$m * N[(b * N[(Pi * N[(0.005555555555555556 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999999e-234

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in angle around 0 83.2%

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

   6. Taylor expanded in angle around 0 75.5%

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

      1. unpow2 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*l* 76.3%

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

      4. associate-*r* 76.3%

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

      5. *-commutative 76.3%

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

      6. associate-*l* 76.3%

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

      7. associate-*r* 76.3%

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

      8. *-commutative 76.3%

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

      9. associate-*r* 76.3%

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

      10. *-commutative 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. associate-*l* 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

   10. Simplified 75.8%

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

   if 2.7999999999999999e-234 < b

   1. Initial program 81.5%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in angle around 0 81.3%

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

   6. Taylor expanded in angle around 0 77.9%

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

      1. unpow2 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-*l* 75.1%

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

      4. associate-*r* 75.1%

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

      5. *-commutative 75.1%

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

      6. associate-*l* 75.1%

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

      7. associate-*r* 75.1%

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

      8. *-commutative 75.1%

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

      9. associate-*r* 75.1%

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

      10. *-commutative 75.1%

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

   8. Applied egg-rr 75.1%

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

      1. associate-*l* 75.1%

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

      2. *-commutative 75.1%

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

      3. *-commutative 75.1%

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

      4. *-commutative 75.1%

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

   10. Simplified 75.1%

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

   11. Taylor expanded in b around 0 75.1%

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

       1. *-commutative 75.1%

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

       2. associate-*l* 75.1%

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

   13. Simplified 75.1%

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

   14. Taylor expanded in angle around 0 75.1%

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

   15. Taylor expanded in angle around 0 64.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   16. Step-by-step derivation

       1. associate-*r* 64.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

       2. unpow2 64.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

       3. unpow2 64.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

       4. unswap-sqr 77.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

       5. *-commutative 77.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left({\pi}^{2} \cdot \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)\right)} \]

       6. unpow2 77.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot \left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)\right) \]

       7. swap-sqr 77.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)} \]

       8. unpow2 77.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

       9. associate-*r* 77.9%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(\pi \cdot angle\right) \cdot b\right)}}^{2} \]

       10. *-commutative 77.9%

           \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\color{blue}{\left(angle \cdot \pi\right)} \cdot b\right)}^{2} \]

       11. associate-*l* 77.9%

           \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(angle \cdot \left(\pi \cdot b\right)\right)}}^{2} \]

   17. Simplified 77.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-234}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot b\right)\\ t_1 := angle\_m \cdot t\_0\\ \mathbf{if}\;b \leq 2 \cdot 10^{-235}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle\_m \cdot \left(b \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + t\_1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 b))) (t_1 (* angle_m t_0)))
   (if (<= b 2e-235)
     (+
      (pow a 2.0)
      (* angle_m (* (* 0.005555555555555556 PI) (* angle_m (* b t_0)))))
     (+ (pow a 2.0) (* t_1 t_1)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * b);
	double t_1 = angle_m * t_0;
	double tmp;
	if (b <= 2e-235) {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (angle_m * (b * t_0))));
	} else {
		tmp = pow(a, 2.0) + (t_1 * t_1);
	}
	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 * b);
	double t_1 = angle_m * t_0;
	double tmp;
	if (b <= 2e-235) {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (angle_m * (b * t_0))));
	} else {
		tmp = Math.pow(a, 2.0) + (t_1 * t_1);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * b)
	t_1 = angle_m * t_0
	tmp = 0
	if b <= 2e-235:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (angle_m * (b * t_0))))
	else:
		tmp = math.pow(a, 2.0) + (t_1 * t_1)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * b))
	t_1 = Float64(angle_m * t_0)
	tmp = 0.0
	if (b <= 2e-235)
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(angle_m * Float64(b * t_0)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(t_1 * t_1));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * b);
	t_1 = angle_m * t_0;
	tmp = 0.0;
	if (b <= 2e-235)
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (angle_m * (b * t_0))));
	else
		tmp = (a ^ 2.0) + (t_1 * t_1);
	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[(0.005555555555555556 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * t$95$0), $MachinePrecision]}, If[LessEqual[b, 2e-235], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(angle$95$m * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999999e-235

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in angle around 0 83.2%

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

   6. Taylor expanded in angle around 0 75.5%

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

      1. unpow2 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*l* 76.3%

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

      4. associate-*r* 76.3%

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

      5. *-commutative 76.3%

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

      6. associate-*l* 76.3%

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

      7. associate-*r* 76.3%

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

      8. *-commutative 76.3%

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

      9. associate-*r* 76.3%

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

      10. *-commutative 76.3%

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

   8. Applied egg-rr 76.3%

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

      1. associate-*l* 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

   10. Simplified 75.8%

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

   if 1.9999999999999999e-235 < b

   1. Initial program 81.5%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in angle around 0 81.3%

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

   6. Taylor expanded in angle around 0 77.9%

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

      1. unpow2 77.9%

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

      2. associate-*r* 77.9%

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

      3. associate-*r* 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. *-commutative 77.8%

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

      7. *-commutative 77.8%

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

      8. associate-*r* 77.8%

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

      9. *-commutative 77.8%

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

   8. Applied egg-rr 77.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-235}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right) \cdot \left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot b\right)\\ \mathbf{if}\;b \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle\_m \cdot \left(b \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(angle\_m \cdot \left(t\_0 \cdot \frac{angle\_m}{\frac{180}{\pi}}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 b))))
   (if (<= b 3.8e-222)
     (+
      (pow a 2.0)
      (* angle_m (* (* 0.005555555555555556 PI) (* angle_m (* b t_0)))))
     (+ (pow a 2.0) (* b (* angle_m (* t_0 (/ angle_m (/ 180.0 PI)))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * b);
	double tmp;
	if (b <= 3.8e-222) {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (angle_m * (b * t_0))));
	} else {
		tmp = pow(a, 2.0) + (b * (angle_m * (t_0 * (angle_m / (180.0 / ((double) M_PI))))));
	}
	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 * b);
	double tmp;
	if (b <= 3.8e-222) {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (angle_m * (b * t_0))));
	} else {
		tmp = Math.pow(a, 2.0) + (b * (angle_m * (t_0 * (angle_m / (180.0 / Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * b)
	tmp = 0
	if b <= 3.8e-222:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (angle_m * (b * t_0))))
	else:
		tmp = math.pow(a, 2.0) + (b * (angle_m * (t_0 * (angle_m / (180.0 / math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * b))
	tmp = 0.0
	if (b <= 3.8e-222)
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(angle_m * Float64(b * t_0)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(b * Float64(angle_m * Float64(t_0 * Float64(angle_m / Float64(180.0 / pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * b);
	tmp = 0.0;
	if (b <= 3.8e-222)
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (angle_m * (b * t_0))));
	else
		tmp = (a ^ 2.0) + (b * (angle_m * (t_0 * (angle_m / (180.0 / pi)))));
	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[(0.005555555555555556 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.8e-222], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(angle$95$m * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(b * N[(angle$95$m * N[(t$95$0 * N[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.79999999999999997e-222

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in angle around 0 82.8%

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

   6. Taylor expanded in angle around 0 75.2%

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

      1. unpow2 75.2%

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

      2. *-commutative 75.2%

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

      3. associate-*l* 75.9%

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

      4. associate-*r* 75.9%

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

      5. *-commutative 75.9%

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

      6. associate-*l* 75.9%

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

      7. associate-*r* 75.9%

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

      8. *-commutative 75.9%

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

      9. associate-*r* 75.9%

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

      10. *-commutative 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. associate-*l* 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r* 75.4%

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

      4. *-commutative 75.4%

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

      5. *-commutative 75.4%

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

   10. Simplified 75.4%

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

   if 3.79999999999999997e-222 < b

   1. Initial program 82.1%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in angle around 0 82.0%

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

   6. Taylor expanded in angle around 0 78.5%

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

      1. unpow2 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*l* 75.7%

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

      4. associate-*r* 75.7%

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

      5. *-commutative 75.7%

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

      6. associate-*l* 75.7%

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

      7. associate-*r* 75.7%

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

      8. *-commutative 75.7%

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

      9. associate-*r* 75.7%

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

      10. *-commutative 75.7%

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

   8. Applied egg-rr 75.7%

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

      1. associate-*r* 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*r* 76.6%

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

      4. associate-*r* 76.6%

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

      5. *-commutative 76.6%

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

      6. *-commutative 76.6%

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

      7. metadata-eval 76.6%

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

      8. associate-/r/ 76.6%

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

      9. associate-*l/ 76.6%

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

      10. *-lft-identity 76.6%

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

      11. *-commutative 76.6%

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

   10. Simplified 76.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{b \cdot \left(angle \cdot \left(\frac{angle}{\frac{180}{\pi}} \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(angle \cdot \left(\left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right) \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot b\right)\\ \mathbf{if}\;angle\_m \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(b \cdot \left(angle\_m \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle\_m \cdot \left(b \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 b))))
   (if (<= angle_m 4.5e-73)
     (+
      (pow a 2.0)
      (* angle_m (* (* 0.005555555555555556 PI) (* b (* angle_m t_0)))))
     (+
      (pow a 2.0)
      (* angle_m (* (* 0.005555555555555556 PI) (* angle_m (* b t_0))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * b);
	double tmp;
	if (angle_m <= 4.5e-73) {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (b * (angle_m * t_0))));
	} else {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (angle_m * (b * t_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 * b);
	double tmp;
	if (angle_m <= 4.5e-73) {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (b * (angle_m * t_0))));
	} else {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (angle_m * (b * t_0))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * b)
	tmp = 0
	if angle_m <= 4.5e-73:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (b * (angle_m * t_0))))
	else:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (angle_m * (b * t_0))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * b))
	tmp = 0.0
	if (angle_m <= 4.5e-73)
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(b * Float64(angle_m * t_0)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(angle_m * Float64(b * t_0)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * b);
	tmp = 0.0;
	if (angle_m <= 4.5e-73)
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (b * (angle_m * t_0))));
	else
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (angle_m * (b * t_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[(0.005555555555555556 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle$95$m, 4.5e-73], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(b * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(angle$95$m * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.5e-73

   1. Initial program 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in angle around 0 88.2%

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

   6. Taylor expanded in angle around 0 83.9%

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

      1. unpow2 83.9%

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

      2. *-commutative 83.9%

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

      3. associate-*l* 82.6%

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

      4. associate-*r* 82.6%

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

      5. *-commutative 82.6%

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

      6. associate-*l* 82.6%

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

      7. associate-*r* 82.6%

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

      8. *-commutative 82.6%

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

      9. associate-*r* 82.6%

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

      10. *-commutative 82.6%

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

   8. Applied egg-rr 82.6%

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

      1. associate-*l* 82.6%

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

      2. *-commutative 82.6%

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

      3. *-commutative 82.6%

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

      4. *-commutative 82.6%

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

   10. Simplified 82.6%

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

   if 4.5e-73 < angle

   1. Initial program 68.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in angle around 0 68.6%

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

   6. Taylor expanded in angle around 0 58.0%

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

      1. unpow2 58.0%

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

      2. *-commutative 58.0%

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

      3. associate-*l* 59.2%

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

      4. associate-*r* 59.2%

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

      5. *-commutative 59.2%

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

      6. associate-*l* 59.2%

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

      7. associate-*r* 59.2%

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

      8. *-commutative 59.2%

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

      9. associate-*r* 59.2%

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

      10. *-commutative 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. associate-*l* 59.2%

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

      2. *-commutative 59.2%

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

      3. associate-*r* 63.4%

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

      4. *-commutative 63.4%

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

      5. *-commutative 63.4%

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

   10. Simplified 63.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4.5 \cdot 10^{-73}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(b \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle\_m \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle\_m \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 2.1e-141)
   (+
    (pow a 2.0)
    (*
     angle_m
     (*
      (* 0.005555555555555556 PI)
      (* b (* 0.005555555555555556 (* angle_m (* PI b)))))))
   (+
    (pow a 2.0)
    (*
     angle_m
     (*
      (* 0.005555555555555556 PI)
      (* angle_m (* b (* PI (* 0.005555555555555556 b)))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.1e-141) {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (b * (0.005555555555555556 * (angle_m * (((double) M_PI) * b))))));
	} else {
		tmp = pow(a, 2.0) + (angle_m * ((0.005555555555555556 * ((double) M_PI)) * (angle_m * (b * (((double) M_PI) * (0.005555555555555556 * b))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.1e-141) {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (b * (0.005555555555555556 * (angle_m * (Math.PI * b))))));
	} else {
		tmp = Math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * Math.PI) * (angle_m * (b * (Math.PI * (0.005555555555555556 * b))))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if angle_m <= 2.1e-141:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (b * (0.005555555555555556 * (angle_m * (math.pi * b))))))
	else:
		tmp = math.pow(a, 2.0) + (angle_m * ((0.005555555555555556 * math.pi) * (angle_m * (b * (math.pi * (0.005555555555555556 * b))))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 2.1e-141)
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(b * Float64(0.005555555555555556 * Float64(angle_m * Float64(pi * b)))))));
	else
		tmp = Float64((a ^ 2.0) + Float64(angle_m * Float64(Float64(0.005555555555555556 * pi) * Float64(angle_m * Float64(b * Float64(pi * Float64(0.005555555555555556 * b)))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 2.1e-141)
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (b * (0.005555555555555556 * (angle_m * (pi * b))))));
	else
		tmp = (a ^ 2.0) + (angle_m * ((0.005555555555555556 * pi) * (angle_m * (b * (pi * (0.005555555555555556 * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 2.1e-141], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(b * N[(0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(angle$95$m * N[(b * N[(Pi * N[(0.005555555555555556 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.0999999999999999e-141

   1. Initial program 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in angle around 0 87.6%

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

   6. Taylor expanded in angle around 0 83.1%

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

      1. unpow2 83.1%

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

      2. *-commutative 83.1%

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

      3. associate-*l* 82.2%

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

      4. associate-*r* 82.2%

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

      5. *-commutative 82.2%

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

      6. associate-*l* 82.2%

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

      7. associate-*r* 82.2%

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

      8. *-commutative 82.2%

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

      9. associate-*r* 82.2%

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

      10. *-commutative 82.2%

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

   8. Applied egg-rr 82.2%

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

      1. associate-*l* 82.2%

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

      2. *-commutative 82.2%

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

      3. *-commutative 82.2%

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

      4. *-commutative 82.2%

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

   10. Simplified 82.2%

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

   11. Taylor expanded in b around 0 82.2%

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

       1. *-commutative 82.2%

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

       2. associate-*l* 82.2%

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

   13. Simplified 82.2%

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

   14. Taylor expanded in angle around 0 82.2%

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

   if 2.0999999999999999e-141 < angle

   1. Initial program 72.3%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in angle around 0 72.0%

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

   6. Taylor expanded in angle around 0 62.5%

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

      1. unpow2 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-*l* 62.6%

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

      4. associate-*r* 62.6%

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

      5. *-commutative 62.6%

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

      6. associate-*l* 62.6%

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

      7. associate-*r* 62.6%

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

      8. *-commutative 62.6%

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

      9. associate-*r* 62.6%

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

      10. *-commutative 62.6%

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

   8. Applied egg-rr 62.6%

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

      1. associate-*l* 62.6%

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

      2. *-commutative 62.6%

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

      3. associate-*r* 66.3%

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

      4. *-commutative 66.3%

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

      5. *-commutative 66.3%

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

   10. Simplified 66.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Taylor expanded in angle around 0 76.4%

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

   1. unpow2 76.4%

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

   2. *-commutative 76.4%

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

   3. associate-*l* 75.8%

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

   4. associate-*r* 75.8%

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

   5. *-commutative 75.8%

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

   6. associate-*l* 75.8%

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

   7. associate-*r* 75.8%

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

   8. *-commutative 75.8%

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

   9. associate-*r* 75.8%

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

   10. *-commutative 75.8%

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

8. Applied egg-rr 75.8%

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

   1. associate-*l* 75.8%

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

   2. *-commutative 75.8%

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

   3. associate-*r* 72.9%

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

   4. *-commutative 72.9%

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

   5. *-commutative 72.9%

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

10. Simplified 72.9%

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

11. Final simplification 72.9%

    \[\leadsto {a}^{2} + angle \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))