ab-angle->ABCF C

Percentage Accurate: 80.0% → 79.9%

Time: 54.3s

Alternatives: 17

Speedup: 1.0×

• 36.4% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. metadata-eval 76.9%

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

   2. div-inv 77.0%

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

   3. add-cube-cbrt 77.0%

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

   4. pow3 77.0%

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

   5. div-inv 77.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. metadata-eval 77.0%

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

6. Applied egg-rr 77.0%

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

   1. cube-mult 77.1%

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

   2. add-sqr-sqrt 40.2%

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

   3. associate-*l* 40.3%

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

   4. pow1/3 40.1%

      \[\leadsto {\left(a \cdot \cos \left(\sqrt{\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. sqrt-pow1 40.1%

      \[\leadsto {\left(a \cdot \cos \left(\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 40.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot \left(\sqrt{\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. metadata-eval 40.1%

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

   8. pow1/3 40.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt{\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{0.3333333333333333}}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. sqrt-pow1 40.2%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. *-commutative 40.2%

       \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot \left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   11. metadata-eval 40.2%

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

   12. pow2 40.2%

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

8. Applied egg-rr 31.1%

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

   1. *-un-lft-identity 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\left(1 \cdot \sqrt[3]{{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. *-commutative 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\left(\sqrt[3]{{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2}} \cdot 1\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. *-commutative 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt[3]{{\color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{2}} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. associate-*r* 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt[3]{{\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}^{2}} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow2 31.1%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\sqrt[3]{\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. cbrt-prod 40.3%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{\left(\sqrt[3]{0.005555555555555556 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. pow2 40.3%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left(\color{blue}{{\left(\sqrt[3]{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right)}^{2}} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. *-commutative 40.3%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}}\right)}^{2} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. *-commutative 40.3%

      \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.005555555555555556}\right)}^{2} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. associate-*r* 40.2%

       \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}}\right)}^{2} \cdot 1\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 40.2%

    \[\leadsto {\left(a \cdot \cos \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \left({\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{2} \cdot 1\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
11. Step-by-step derivation

    1. *-rgt-identity 40.2%

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

    2. *-commutative 40.2%

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

    3. *-commutative 40.2%

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

    4. associate-*r* 40.3%

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

12. Simplified 40.3%

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

13. Final simplification 40.3%

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

Alternative 2: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ t_1 := t\_0 + 1\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\sin t\_1, \sin 1, \cos 1 \cdot \cos t\_1\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m))) (t_1 (+ t_0 1.0)))
   (+
    (pow (* b (sin t_0)) 2.0)
    (pow (* a (fma (sin t_1) (sin 1.0) (* (cos 1.0) (cos t_1)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double t_1 = t_0 + 1.0;
	return pow((b * sin(t_0)), 2.0) + pow((a * fma(sin(t_1), sin(1.0), (cos(1.0) * cos(t_1)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	t_1 = Float64(t_0 + 1.0)
	return Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * fma(sin(t_1), sin(1.0), Float64(cos(1.0) * cos(t_1)))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. metadata-eval 76.9%

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

   2. div-inv 77.0%

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

   3. expm1-log1p-u 63.6%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. expm1-undefine 63.6%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. cos-diff 63.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. div-inv 63.5%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. metadata-eval 63.5%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 63.5%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 63.5%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 63.5%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. +-commutative 63.5%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1 + \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. fma-define 63.5%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{fma}\left(\sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. log1p-undefine 63.5%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(e^{\color{blue}{\log \left(1 + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. rem-exp-log 63.5%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \color{blue}{\left(1 + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}, \sin 1, \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. *-commutative 63.5%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 63.5%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), \sin 1, \color{blue}{\cos 1 \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log1p-undefine 63.5%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), \sin 1, \cos 1 \cdot \cos \left(e^{\color{blue}{\log \left(1 + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. rem-exp-log 77.0%

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

   9. *-commutative 77.0%

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

8. Simplified 77.0%

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

9. Final simplification 77.0%

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

Alternative 3: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos \left(e^{\left(3 \cdot \log t\_0\right) \cdot 0.3333333333333333}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m))))
   (+
    (pow (* b (sin t_0)) 2.0)
    (pow (* a (cos (exp (* (* 3.0 (log t_0)) 0.3333333333333333)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	return pow((b * sin(t_0)), 2.0) + pow((a * cos(exp(((3.0 * log(t_0)) * 0.3333333333333333)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle_m);
	return Math.pow((b * Math.sin(t_0)), 2.0) + Math.pow((a * Math.cos(Math.exp(((3.0 * Math.log(t_0)) * 0.3333333333333333)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * angle_m)
	return math.pow((b * math.sin(t_0)), 2.0) + math.pow((a * math.cos(math.exp(((3.0 * math.log(t_0)) * 0.3333333333333333)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	return Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * cos(exp(Float64(Float64(3.0 * log(t_0)) * 0.3333333333333333)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * angle_m);
	tmp = ((b * sin(t_0)) ^ 2.0) + ((a * cos(exp(((3.0 * log(t_0)) * 0.3333333333333333)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[Exp[N[(N[(3.0 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. metadata-eval 76.9%

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

   2. div-inv 77.0%

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

   3. add-cbrt-cube 58.1%

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

   4. pow1/3 46.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}^{0.3333333333333333}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow-to-exp 46.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 0.3333333333333333}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow3 46.1%

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left({\left(\pi \cdot \frac{angle}{180}\right)}^{3}\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log-pow 40.3%

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\left(3 \cdot \log \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 40.3%

      \[\leadsto {\left(a \cdot \cos \left(e^{\left(3 \cdot \log \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 40.3%

      \[\leadsto {\left(a \cdot \cos \left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 40.3%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.3333333333333333}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

7. Final simplification 40.3%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(e^{\left(3 \cdot \log \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 4: 80.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. add-exp-log 40.2%

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

6. Applied egg-rr 40.2%

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

7. Final simplification 40.2%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((a * cos((1.0 / (180.0 / (pi * 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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(1.0 / N[(180.0 / N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. metadata-eval 76.9%

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

   2. div-inv 77.0%

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

   3. associate-*r/ 77.0%

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

   4. clear-num 77.0%

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

6. Applied egg-rr 77.0%

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

7. Final simplification 77.0%

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

Alternative 6: 80.0% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0)
  (pow (* a (cos (* 0.005555555555555556 (/ PI (/ 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) * (0.005555555555555556 * angle_m)))), 2.0) + pow((a * cos((0.005555555555555556 * (((double) M_PI) / (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 * (0.005555555555555556 * angle_m)))), 2.0) + Math.pow((a * Math.cos((0.005555555555555556 * (Math.PI / (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 * (0.005555555555555556 * angle_m)))), 2.0) + math.pow((a * math.cos((0.005555555555555556 * (math.pi / (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(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(a * cos(Float64(0.005555555555555556 * Float64(pi / Float64(1.0 / angle_m))))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((a * cos((0.005555555555555556 * (pi / (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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(Pi / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. metadata-eval 76.9%

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

   2. div-inv 77.0%

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

   3. add-cbrt-cube 58.1%

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

   4. pow1/3 46.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}^{0.3333333333333333}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow-to-exp 46.1%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\log \left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot 0.3333333333333333}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow3 46.1%

      \[\leadsto {\left(a \cdot \cos \left(e^{\log \color{blue}{\left({\left(\pi \cdot \frac{angle}{180}\right)}^{3}\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. log-pow 40.3%

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\left(3 \cdot \log \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. div-inv 40.3%

      \[\leadsto {\left(a \cdot \cos \left(e^{\left(3 \cdot \log \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. metadata-eval 40.3%

      \[\leadsto {\left(a \cdot \cos \left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right) \cdot 0.3333333333333333}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 40.3%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\left(3 \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot 0.3333333333333333}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. *-commutative 40.3%

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. associate-*r* 40.2%

      \[\leadsto {\left(a \cdot \cos \left(e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. metadata-eval 40.2%

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

   4. *-un-lft-identity 40.2%

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

   5. add-exp-log 76.9%

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

   6. metadata-eval 76.9%

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

   7. div-inv 77.0%

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

   8. clear-num 76.9%

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

   9. associate-*r/ 76.9%

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

   10. *-commutative 76.9%

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

   11. div-inv 76.9%

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

   12. times-frac 77.0%

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

   13. metadata-eval 77.0%

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

8. Applied egg-rr 77.0%

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

9. Final simplification 77.0%

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

Alternative 7: 80.0% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0)
  (pow (* a (cos (/ PI (/ 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) * (0.005555555555555556 * angle_m)))), 2.0) + pow((a * cos((((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) {
	return Math.pow((b * Math.sin((Math.PI * (0.005555555555555556 * angle_m)))), 2.0) + Math.pow((a * Math.cos((Math.PI / (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 * (0.005555555555555556 * angle_m)))), 2.0) + math.pow((a * math.cos((math.pi / (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(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(a * cos(Float64(pi / Float64(180.0 / angle_m)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((a * cos((pi / (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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. metadata-eval 76.9%

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

   2. div-inv 77.0%

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

   3. clear-num 76.9%

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

   4. un-div-inv 76.9%

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

6. Applied egg-rr 76.9%

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

7. Final simplification 76.9%

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

Alternative 8: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

   1. metadata-eval 76.9%

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

   2. div-inv 77.0%

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

   3. clear-num 76.9%

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

   4. un-div-inv 76.9%

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

6. Applied egg-rr 76.9%

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

7. Taylor expanded in angle around inf 76.9%

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

8. Final simplification 76.9%

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

Alternative 9: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Final simplification 76.9%

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

Alternative 10: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around inf 76.9%

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

6. Final simplification 76.9%

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

Alternative 11: 80.0% accurate, 1.3× speedup

Math

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

FPCore

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * 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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Final simplification 76.6%

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

Alternative 12: 79.9% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* b (sin (* 0.005555555555555556 (* PI 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((0.005555555555555556 * (((double) M_PI) * 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((0.005555555555555556 * (Math.PI * 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((0.005555555555555556 * (math.pi * 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(0.005555555555555556 * Float64(pi * angle_m)))) ^ 2.0) + (a ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((0.005555555555555556 * (pi * 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[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around inf 76.6%

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

7. Final simplification 76.6%

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

Alternative 13: 75.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around 0 71.5%

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

   1. unpow2 71.5%

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

   2. associate-*r* 71.5%

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

   3. *-commutative 71.5%

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

   4. *-commutative 71.5%

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

   5. associate-*l* 71.0%

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

   6. *-commutative 71.0%

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

   7. associate-*r* 71.0%

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

   8. *-commutative 71.0%

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

   9. associate-*r* 71.0%

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

   10. *-commutative 71.0%

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

8. Applied egg-rr 71.0%

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

   1. associate-*r* 71.0%

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

   2. *-commutative 71.0%

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

   3. associate-*r* 71.0%

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

   4. associate-*l* 71.0%

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

10. Simplified 71.0%

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

    1. expm1-log1p-u 54.5%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right)}\right) \cdot angle\right) \]

    2. expm1-undefine 56.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)} - 1\right)}\right) \cdot angle\right) \]

    3. associate-*r* 56.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(0.005555555555555556 \cdot b\right) \cdot \pi}\right)} - 1\right)\right) \cdot angle\right) \]

    4. *-commutative 56.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.005555555555555556 \cdot b\right)}\right)} - 1\right)\right) \cdot angle\right) \]

12. Applied egg-rr 56.6%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)} - 1\right)}\right) \cdot angle\right) \]
13. Step-by-step derivation

    1. sub-neg 56.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)} + \left(-1\right)\right)}\right) \cdot angle\right) \]

    2. metadata-eval 56.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)} + \color{blue}{-1}\right)\right) \cdot angle\right) \]

    3. +-commutative 56.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + b \cdot \left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right)}\right)}\right) \cdot angle\right) \]

    4. log1p-undefine 56.6%

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

    5. rem-exp-log 73.1%

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

    6. +-commutative 73.1%

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

    7. associate-*r* 73.1%

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

    8. *-commutative 73.1%

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

    9. associate-*l* 73.1%

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

    10. fma-define 73.1%

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

14. Simplified 73.1%

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

15. Final simplification 73.1%

    \[\leadsto {a}^{2} + b \cdot \left(angle \cdot \left(\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(-1 + \mathsf{fma}\left(0.005555555555555556, \pi \cdot b, 1\right)\right)\right)\right) \]
16. Add Preprocessing

Alternative 14: 75.0% accurate, 2.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow a 2.0) (pow (* b (* angle_m (* PI 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 * (angle_m * (((double) M_PI) * 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 * (angle_m * (Math.PI * 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 * (angle_m * (math.pi * 0.005555555555555556))), 2.0)

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a ^ 2.0) + ((b * (angle_m * (pi * 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[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around 0 71.5%

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

   1. associate-*r* 71.5%

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

   2. *-commutative 71.5%

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

   3. associate-*r* 71.5%

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

8. Simplified 71.5%

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

9. Final simplification 71.5%

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

Alternative 15: 74.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a ^ 2.0) + ((angle_m * (0.005555555555555556 * (pi * b))) ^ 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[(angle$95$m * N[(0.005555555555555556 * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around 0 71.5%

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

   1. *-un-lft-identity 71.5%

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

   2. associate-*r* 71.5%

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

   3. *-commutative 71.5%

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

   4. associate-*r* 71.5%

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

   5. *-commutative 71.5%

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

8. Applied egg-rr 71.5%

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

   1. *-lft-identity 71.5%

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

   2. *-commutative 71.5%

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

   3. associate-*l* 71.5%

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

   4. *-commutative 71.5%

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

10. Simplified 71.5%

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

11. Final simplification 71.5%

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

Alternative 16: 75.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around 0 71.5%

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

   1. unpow2 71.5%

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

   2. associate-*r* 71.5%

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

   3. associate-*r* 71.5%

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

   4. *-commutative 71.5%

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

   5. associate-*r* 71.5%

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

   6. *-commutative 71.5%

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

   7. *-commutative 71.5%

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

   8. associate-*r* 71.5%

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

   9. *-commutative 71.5%

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

8. Applied egg-rr 71.5%

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

9. Final simplification 71.5%

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

Alternative 17: 73.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {a}^{2} + angle\_m \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot \left(\pi \cdot \left(angle\_m \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right) \end{array} \]

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(a, 2.0) + (angle_m * (b * (((double) M_PI) * (b * (((double) M_PI) * (angle_m * 3.08641975308642e-5))))));
}

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 * (b * (Math.PI * (b * (Math.PI * (angle_m * 3.08641975308642e-5))))));
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow(a, 2.0) + (angle_m * (b * (math.pi * (b * (math.pi * (angle_m * 3.08641975308642e-5))))))

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((a ^ 2.0) + Float64(angle_m * Float64(b * Float64(pi * Float64(b * Float64(pi * Float64(angle_m * 3.08641975308642e-5)))))))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a ^ 2.0) + (angle_m * (b * (pi * (b * (pi * (angle_m * 3.08641975308642e-5))))));
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[(b * N[(Pi * N[(b * N[(Pi * N[(angle$95$m * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around 0 71.5%

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

   1. unpow2 71.5%

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

   2. associate-*r* 71.5%

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

   3. *-commutative 71.5%

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

   4. *-commutative 71.5%

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

   5. associate-*l* 71.0%

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

   6. *-commutative 71.0%

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

   7. associate-*r* 71.0%

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

   8. *-commutative 71.0%

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

   9. associate-*r* 71.0%

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

   10. *-commutative 71.0%

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

8. Applied egg-rr 71.0%

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

   1. associate-*r* 71.0%

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

   2. *-commutative 71.0%

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

   3. associate-*r* 71.0%

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

   4. associate-*l* 71.0%

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

10. Simplified 71.0%

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

    1. pow1 71.0%

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

    2. *-commutative 71.0%

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

    3. associate-*l* 71.5%

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

    4. associate-*r* 71.5%

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

    5. *-commutative 71.5%

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

    6. *-commutative 71.5%

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

    7. *-commutative 71.5%

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

    8. associate-*l* 71.5%

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

    9. *-commutative 71.5%

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

    10. metadata-eval 71.5%

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

12. Applied egg-rr 71.5%

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

    1. unpow1 71.5%

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

    2. associate-*r* 71.0%

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

    3. *-commutative 71.0%

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

    4. associate-*l* 71.1%

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

    5. associate-*l* 71.1%

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

    6. associate-*l* 71.1%

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

14. Simplified 71.1%

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

15. Final simplification 71.1%

    \[\leadsto {a}^{2} + angle \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot \left(\pi \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(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)))