ab-angle->ABCF A

Percentage Accurate: 80.0% → 79.9%

Time: 59.8s

Alternatives: 15

Speedup: 1.0×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((angle_m / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(pow(sqrt(((double) M_E)), (2.0 * log((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 * Math.sin(((angle_m / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(Math.pow(Math.sqrt(Math.E), (2.0 * Math.log((angle_m * (Math.PI * 0.005555555555555556))))))), 2.0);
}

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(angle_m / 180.0) * pi))) ^ 2.0) + (Float64(b * cos((sqrt(exp(1)) ^ Float64(2.0 * log(Float64(angle_m * Float64(pi * 0.005555555555555556))))))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((angle_m / 180.0) * pi))) ^ 2.0) + ((b * cos((sqrt(2.71828182845904523536) ^ (2.0 * log((angle_m * (pi * 0.005555555555555556))))))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. *-commutative 81.9%

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

   2. clear-num 81.9%

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

   3. un-div-inv 82.0%

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

4. Applied egg-rr 82.0%

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

   1. associate-/r/ 81.9%

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

   2. div-inv 81.9%

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

   3. metadata-eval 81.9%

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

   4. *-commutative 81.9%

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

   5. rem-exp-log 37.5%

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

   6. *-un-lft-identity 37.5%

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

   7. exp-prod 37.6%

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

   8. associate-*r* 37.5%

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

   9. metadata-eval 37.5%

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

   10. div-inv 37.5%

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

   11. associate-*l/ 37.6%

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

   12. *-commutative 37.6%

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

   13. div-inv 37.6%

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

   14. metadata-eval 37.6%

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

6. Applied egg-rr 37.6%

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

   1. exp-1-e 37.6%

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

   2. *-commutative 37.6%

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

8. Simplified 37.6%

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

   1. add-sqr-sqrt 37.6%

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

   2. unpow-prod-down 37.6%

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

10. Applied egg-rr 37.6%

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

    1. pow-sqr 37.7%

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

    2. associate-*r* 37.7%

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

    3. *-commutative 37.7%

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

    4. *-commutative 37.7%

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

12. Simplified 37.7%

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

13. Final simplification 37.7%

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

Alternative 2: 80.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. *-commutative 81.9%

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

   2. clear-num 81.9%

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

   3. un-div-inv 82.0%

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

4. Applied egg-rr 82.0%

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

   1. add-sqr-sqrt 82.0%

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

   2. pow2 82.0%

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

6. Applied egg-rr 82.0%

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

Alternative 3: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((angle_m / 180.0) * ((double) M_PI)))), 2.0) + pow((b * 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((a * Math.sin(((angle_m / 180.0) * Math.PI))), 2.0) + Math.pow((b * 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((a * math.sin(((angle_m / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos((math.pi / (180.0 / angle_m)))), 2.0)

Julia

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

MATLAB

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

Derivation

1. Initial program 81.9%

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

   1. *-commutative 81.9%

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

   2. clear-num 81.9%

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

   3. un-div-inv 82.0%

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

4. Applied egg-rr 82.0%

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

Alternative 4: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. *-commutative 81.9%

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

   2. clear-num 81.9%

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

   3. un-div-inv 82.0%

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

4. Applied egg-rr 82.0%

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

5. Taylor expanded in angle around inf 81.8%

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

   1. associate-*r* 82.0%

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

   2. *-commutative 82.0%

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

   3. *-commutative 82.0%

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

   4. *-commutative 82.0%

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

7. Simplified 82.0%

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

8. Final simplification 82.0%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

Alternative 6: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around inf 81.9%

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

Alternative 7: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around inf 81.7%

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

6. Final simplification 81.7%

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

Alternative 8: 79.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Final simplification 81.3%

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

Alternative 9: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around inf 81.1%

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

7. Final simplification 81.1%

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

Alternative 10: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \frac{\frac{180}{angle\_m \cdot \pi}}{a}\\ \mathbf{if}\;angle\_m \leq 1.75 \cdot 10^{-181}:\\ \;\;\;\;{b}^{2} + \frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right) + -1\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (/ (/ 180.0 (* angle_m PI)) a)))
   (if (<= angle_m 1.75e-181)
     (+ (pow b 2.0) (/ 1.0 (* t_0 t_0)))
     (+
      (pow b 2.0)
      (pow
       (+ (+ 1.0 (* 0.005555555555555556 (* a (* angle_m PI)))) -1.0)
       2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (180.0 / (angle_m * ((double) M_PI))) / a;
	double tmp;
	if (angle_m <= 1.75e-181) {
		tmp = pow(b, 2.0) + (1.0 / (t_0 * t_0));
	} else {
		tmp = pow(b, 2.0) + pow(((1.0 + (0.005555555555555556 * (a * (angle_m * ((double) M_PI))))) + -1.0), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (180.0 / (angle_m * Math.PI)) / a;
	double tmp;
	if (angle_m <= 1.75e-181) {
		tmp = Math.pow(b, 2.0) + (1.0 / (t_0 * t_0));
	} else {
		tmp = Math.pow(b, 2.0) + Math.pow(((1.0 + (0.005555555555555556 * (a * (angle_m * Math.PI)))) + -1.0), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (180.0 / (angle_m * math.pi)) / a
	tmp = 0
	if angle_m <= 1.75e-181:
		tmp = math.pow(b, 2.0) + (1.0 / (t_0 * t_0))
	else:
		tmp = math.pow(b, 2.0) + math.pow(((1.0 + (0.005555555555555556 * (a * (angle_m * math.pi)))) + -1.0), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(180.0 / Float64(angle_m * pi)) / a)
	tmp = 0.0
	if (angle_m <= 1.75e-181)
		tmp = Float64((b ^ 2.0) + Float64(1.0 / Float64(t_0 * t_0)));
	else
		tmp = Float64((b ^ 2.0) + (Float64(Float64(1.0 + Float64(0.005555555555555556 * Float64(a * Float64(angle_m * pi)))) + -1.0) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = (180.0 / (angle_m * pi)) / a;
	tmp = 0.0;
	if (angle_m <= 1.75e-181)
		tmp = (b ^ 2.0) + (1.0 / (t_0 * t_0));
	else
		tmp = (b ^ 2.0) + (((1.0 + (0.005555555555555556 * (a * (angle_m * pi)))) + -1.0) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[angle$95$m, 1.75e-181], N[(N[Power[b, 2.0], $MachinePrecision] + N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(N[(1.0 + N[(0.005555555555555556 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.74999999999999998e-181

   1. Initial program 86.6%

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

      1. associate-*l/ 86.4%

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

      2. associate-/l* 86.6%

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

      3. cos-neg 86.6%

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

      4. distribute-lft-neg-out 86.6%

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

      5. distribute-frac-neg 86.6%

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

      6. distribute-frac-neg 86.6%

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

      7. distribute-lft-neg-out 86.6%

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

      8. cos-neg 86.6%

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

      9. associate-*l/ 86.7%

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

      10. associate-/l* 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in angle around 0 85.6%

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

   6. Taylor expanded in angle around 0 81.8%

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

      1. *-commutative 81.8%

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

   8. Simplified 81.8%

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

      1. unpow2 81.8%

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

      2. *-commutative 81.8%

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

      3. *-commutative 81.8%

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

      4. associate-*l* 81.8%

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

      5. *-commutative 81.8%

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

      6. metadata-eval 81.8%

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

      7. associate-/r/ 81.9%

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

      8. associate-/r* 81.9%

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

      9. *-commutative 81.9%

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

      10. associate-*l* 80.8%

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

      11. associate-/r* 80.8%

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

      12. associate-/r/ 80.8%

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

      13. metadata-eval 80.8%

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

      14. *-commutative 80.8%

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

      15. associate-*l* 80.9%

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

      16. *-commutative 80.9%

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

      17. associate-*l* 80.9%

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

      18. *-commutative 80.9%

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

   10. Applied egg-rr 80.9%

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

   11. Taylor expanded in angle around 0 80.9%

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

       1. *-commutative 80.9%

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

   13. Simplified 80.9%

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

   14. Taylor expanded in angle around 0 80.9%

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

       1. *-commutative 80.9%

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

       2. *-commutative 80.9%

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

       3. associate-*r* 80.9%

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

       4. associate-*r* 80.9%

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

       5. *-commutative 80.9%

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

       6. associate-*l* 80.9%

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

       7. *-commutative 80.9%

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

   16. Simplified 80.9%

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

       1. associate-*r* 80.9%

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

       2. *-commutative 80.9%

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

       3. metadata-eval 80.9%

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

       4. metadata-eval 80.9%

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

       5. metadata-eval 80.9%

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

       6. sqrt-pow2 80.8%

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

       7. div-inv 80.8%

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

       8. associate-/r/ 80.8%

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

       9. sqrt-pow2 80.9%

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

       10. metadata-eval 80.9%

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

       11. metadata-eval 80.9%

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

       12. clear-num 80.9%

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

       13. associate-*l/ 80.9%

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

       14. *-un-lft-identity 80.9%

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

       15. metadata-eval 80.9%

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

       16. metadata-eval 80.9%

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

       17. sqrt-pow2 80.8%

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

       18. associate-/l/ 80.8%

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

       19. *-commutative 80.8%

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

       20. sqrt-pow2 80.9%

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

       21. metadata-eval 80.9%

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

       22. metadata-eval 80.9%

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

   18. Applied egg-rr 80.9%

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

       1. associate-*r* 80.8%

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

       2. *-commutative 80.8%

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

       3. *-commutative 80.8%

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

       4. *-commutative 80.8%

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

       5. associate-*r* 81.9%

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

       6. *-commutative 81.9%

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

       7. metadata-eval 81.9%

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

       8. div-inv 81.9%

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

       9. clear-num 81.9%

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

       10. un-div-inv 81.9%

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

       11. clear-num 81.9%

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

       12. metadata-eval 81.9%

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

       13. clear-num 81.9%

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

       14. metadata-eval 81.9%

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

       15. frac-times 81.9%

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

   20. Applied egg-rr 81.9%

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

   if 1.74999999999999998e-181 < angle

   1. Initial program 72.7%

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

      1. associate-*l/ 72.3%

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

      2. associate-/l* 72.7%

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

      3. cos-neg 72.7%

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

      4. distribute-lft-neg-out 72.7%

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

      5. distribute-frac-neg 72.7%

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

      6. distribute-frac-neg 72.7%

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

      7. distribute-lft-neg-out 72.7%

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

      8. cos-neg 72.7%

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

      9. associate-*l/ 72.4%

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

      10. associate-/l* 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in angle around 0 73.1%

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

      1. expm1-log1p-u 58.6%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      2. expm1-undefine 56.9%

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

      3. *-commutative 56.9%

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

      4. div-inv 56.9%

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

      5. metadata-eval 56.9%

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

   7. Applied egg-rr 56.9%

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

   8. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

   10. Simplified 67.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.75 \cdot 10^{-181}:\\ \;\;\;\;{b}^{2} + \frac{1}{\frac{\frac{180}{angle \cdot \pi}}{a} \cdot \frac{\frac{180}{angle \cdot \pi}}{a}}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(\left(1 + 0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) + -1\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 75.9%

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

   1. *-commutative 75.9%

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

8. Simplified 75.9%

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

   1. unpow2 75.9%

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

   2. associate-*r* 75.9%

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

   3. *-commutative 75.9%

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

   4. associate-*l* 75.9%

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

   5. *-commutative 75.9%

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

   6. *-commutative 75.9%

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

   7. associate-*l* 75.9%

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

10. Applied egg-rr 75.9%

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

11. Final simplification 75.9%

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

Alternative 12: 75.0% accurate, 3.5× speedup

Math

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

FPCore

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * Float64(a * 0.005555555555555556))
	return Float64((b ^ 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) * (a * 0.005555555555555556);
	tmp = (b ^ 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[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 75.9%

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

   1. *-commutative 75.9%

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

8. Simplified 75.9%

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

   1. unpow2 75.9%

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

   2. *-commutative 75.9%

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

   3. associate-*l* 75.9%

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

   4. *-commutative 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-*l* 75.9%

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

   7. *-commutative 75.9%

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

10. Applied egg-rr 75.9%

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

11. Final simplification 75.9%

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

Alternative 13: 73.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 75.9%

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

   1. *-commutative 75.9%

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

8. Simplified 75.9%

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

   1. unpow2 75.9%

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

   2. *-commutative 75.9%

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

   3. *-commutative 75.9%

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

   4. associate-*l* 75.9%

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

   5. *-commutative 75.9%

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

   6. metadata-eval 75.9%

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

   7. associate-/r/ 75.9%

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

   8. associate-/r* 75.9%

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

   9. *-commutative 75.9%

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

   10. associate-*l* 75.6%

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

   11. associate-/r* 75.6%

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

   12. associate-/r/ 75.5%

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

   13. metadata-eval 75.5%

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

   14. *-commutative 75.5%

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

   15. associate-*l* 75.6%

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

   16. *-commutative 75.6%

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

   17. associate-*l* 75.6%

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

   18. *-commutative 75.6%

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

10. Applied egg-rr 75.6%

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

11. Final simplification 75.6%

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

Alternative 14: 73.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 75.9%

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

   1. *-commutative 75.9%

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

8. Simplified 75.9%

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

   1. unpow2 75.9%

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

   2. *-commutative 75.9%

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

   3. *-commutative 75.9%

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

   4. associate-*l* 75.9%

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

   5. *-commutative 75.9%

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

   6. metadata-eval 75.9%

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

   7. associate-/r/ 75.9%

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

   8. associate-/r* 75.9%

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

   9. *-commutative 75.9%

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

   10. associate-*l* 75.6%

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

   11. associate-/r* 75.6%

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

   12. associate-/r/ 75.5%

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

   13. metadata-eval 75.5%

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

   14. *-commutative 75.5%

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

   15. associate-*l* 75.6%

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

   16. *-commutative 75.6%

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

   17. associate-*l* 75.6%

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

   18. *-commutative 75.6%

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

10. Applied egg-rr 75.6%

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

11. Taylor expanded in angle around 0 75.6%

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

    1. *-commutative 75.6%

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

13. Simplified 75.6%

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

14. Taylor expanded in angle around 0 75.6%

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

    1. *-commutative 75.6%

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

    2. *-commutative 75.6%

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

    3. associate-*l* 75.6%

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

    4. associate-*l* 75.6%

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

16. Simplified 75.6%

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

17. Taylor expanded in a around 0 75.6%

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

    1. *-commutative 75.6%

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

    2. *-commutative 75.6%

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

    3. associate-*r* 75.6%

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

    4. associate-*r* 75.6%

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

    5. *-commutative 75.6%

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

    6. associate-*l* 75.6%

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

19. Simplified 75.6%

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

20. Final simplification 75.6%

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

Alternative 15: 73.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l/ 81.6%

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

   2. associate-/l* 81.9%

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

   3. cos-neg 81.9%

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

   4. distribute-lft-neg-out 81.9%

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

   5. distribute-frac-neg 81.9%

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

   6. distribute-frac-neg 81.9%

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

   7. distribute-lft-neg-out 81.9%

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

   8. cos-neg 81.9%

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

   9. associate-*l/ 81.8%

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

   10. associate-/l* 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in angle around 0 81.3%

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

6. Taylor expanded in angle around 0 75.9%

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

   1. *-commutative 75.9%

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

8. Simplified 75.9%

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

   1. unpow2 75.9%

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

   2. *-commutative 75.9%

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

   3. *-commutative 75.9%

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

   4. associate-*l* 75.9%

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

   5. *-commutative 75.9%

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

   6. metadata-eval 75.9%

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

   7. associate-/r/ 75.9%

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

   8. associate-/r* 75.9%

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

   9. *-commutative 75.9%

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

   10. associate-*l* 75.6%

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

   11. associate-/r* 75.6%

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

   12. associate-/r/ 75.5%

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

   13. metadata-eval 75.5%

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

   14. *-commutative 75.5%

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

   15. associate-*l* 75.6%

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

   16. *-commutative 75.6%

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

   17. associate-*l* 75.6%

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

   18. *-commutative 75.6%

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

10. Applied egg-rr 75.6%

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

11. Taylor expanded in angle around 0 75.6%

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

    1. *-commutative 75.6%

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

13. Simplified 75.6%

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

14. Final simplification 75.6%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))