ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.2%

Time: 25.1s

Alternatives: 17

Speedup: 1.3×

• 35.9% 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: 80.2% 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: 80.2% accurate, 0.5× speedup

Math

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

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 (* (/ (sqrt angle_m) 180.0) (pow (sqrt (* PI (sqrt angle_m))) 2.0))))
   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(((sqrt(angle_m) / 180.0) * pow(sqrt((((double) M_PI) * sqrt(angle_m))), 2.0)))), 2.0);
}

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. add-sqr-sqrt 40.1%

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

   2. div-inv 40.1%

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

   3. times-frac 40.1%

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

6. Applied egg-rr 40.1%

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

   1. add-sqr-sqrt 40.1%

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

   2. pow2 40.1%

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

   3. associate-/r/ 40.1%

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

   4. sqrt-prod 40.1%

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

   5. /-rgt-identity 40.1%

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

   6. sqrt-prod 40.1%

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

   7. *-commutative 40.1%

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

8. Applied egg-rr 40.1%

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

9. Final simplification 40.1%

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

Alternative 2: 80.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. associate-/r/ 77.5%

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

   2. add-sqr-sqrt 77.4%

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

   3. associate-*r* 77.4%

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

   4. div-inv 77.5%

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

   5. metadata-eval 77.5%

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

6. Applied egg-rr 77.5%

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

7. Final simplification 77.5%

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

Alternative 3: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. clear-num 77.5%

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

   2. associate-/r/ 77.5%

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

   3. clear-num 77.5%

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

   4. add-sqr-sqrt 40.0%

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

   5. associate-*r* 40.0%

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

   6. div-inv 40.0%

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

   7. metadata-eval 40.0%

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

6. Applied egg-rr 40.0%

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

7. Final simplification 40.0%

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

Alternative 4: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.5%

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

   1. *-commutative 77.5%

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

   2. clear-num 77.5%

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

   3. div-inv 77.5%

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

   4. add-sqr-sqrt 77.5%

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

   5. associate-/l* 77.5%

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

4. Applied egg-rr 77.5%

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

5. Final simplification 77.5%

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

Alternative 5: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(angle_m / Float64(180.0 / pi)))) ^ 2.0) + (Float64(b * cos(Float64(sqrt(pi) * Float64(Float64(angle_m * 0.005555555555555556) * sqrt(pi))))) ^ 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) * ((angle_m * 0.005555555555555556) * sqrt(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[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. associate-/r/ 77.5%

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

   2. add-sqr-sqrt 77.4%

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

   3. associate-*r* 77.4%

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

   4. div-inv 77.5%

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

   5. metadata-eval 77.5%

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

6. Applied egg-rr 77.6%

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

7. Final simplification 77.6%

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

Alternative 6: 80.2% accurate, 0.7× speedup

Math

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

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 (* (/ (sqrt angle_m) 180.0) (* PI (sqrt 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(((sqrt(angle_m) / 180.0) * (((double) M_PI) * sqrt(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.sqrt(angle_m) / 180.0) * (Math.PI * Math.sqrt(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.sqrt(angle_m) / 180.0) * (math.pi * math.sqrt(angle_m))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(angle_m / Float64(180.0 / pi)))) ^ 2.0) + (Float64(b * cos(Float64(Float64(sqrt(angle_m) / 180.0) * Float64(pi * sqrt(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(angle_m) / 180.0) * (pi * sqrt(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[(angle$95$m / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(N[Sqrt[angle$95$m], $MachinePrecision] / 180.0), $MachinePrecision] * N[(Pi * N[Sqrt[angle$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. add-sqr-sqrt 40.1%

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

   2. div-inv 40.1%

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

   3. times-frac 40.1%

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

6. Applied egg-rr 40.1%

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

   1. associate-/r/ 40.1%

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

   2. /-rgt-identity 40.1%

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

8. Applied egg-rr 40.1%

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

9. Final simplification 40.1%

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

Alternative 7: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. associate-/r/ 77.5%

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

   2. *-commutative 77.5%

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

   3. clear-num 77.5%

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

   4. div-inv 77.5%

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

   5. frac-2neg 77.5%

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

   6. div-inv 77.5%

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

   7. distribute-neg-frac 77.5%

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

   8. metadata-eval 77.5%

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

6. Applied egg-rr 77.5%

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

7. Final simplification 77.5%

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

Alternative 8: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. associate-/r/ 77.5%

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

   2. *-commutative 77.5%

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

   3. clear-num 77.5%

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

   4. div-inv 77.5%

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

   5. frac-2neg 77.5%

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

   6. div-inv 77.5%

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

   7. distribute-neg-frac 77.5%

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

   8. metadata-eval 77.5%

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

6. Applied egg-rr 77.5%

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

7. Final simplification 77.5%

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

Alternative 9: 80.2% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (cos (/ angle_m (/ 180.0 PI)))) 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((angle_m / (180.0 / ((double) M_PI))))), 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((angle_m / (180.0 / Math.PI)))), 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((angle_m / (180.0 / math.pi)))), 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(angle_m / Float64(180.0 / pi)))) ^ 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((angle_m / (180.0 / pi)))) ^ 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[(angle$95$m / N[(180.0 / Pi), $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 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. associate-/r/ 77.5%

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

   2. div-inv 77.5%

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

   3. metadata-eval 77.5%

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

6. Applied egg-rr 77.5%

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

7. Final simplification 77.5%

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

Alternative 10: 80.2% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 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 = ((double) M_PI) * (angle_m / 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 = Math.PI * (angle_m / 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 = math.pi * (angle_m / 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(pi * Float64(angle_m / 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 = pi * (angle_m / 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[(Pi * N[(angle$95$m / 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 77.5%

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

3. Final simplification 77.5%

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

Alternative 11: 80.2% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (/ angle_m (/ 180.0 PI))))
   (+ (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 / (180.0 / ((double) M_PI));
	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 / (180.0 / Math.PI);
	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 / (180.0 / math.pi)
	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(180.0 / pi))
	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 / (180.0 / pi);
	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[(180.0 / Pi), $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 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

5. Final simplification 77.5%

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

Alternative 12: 80.2% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow b 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, 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, 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, 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((b ^ 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 ^ 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[b, 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 77.5%

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

3. Taylor expanded in angle around 0 77.4%

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

4. Taylor expanded in angle around inf 77.4%

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

5. Final simplification 77.4%

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

Alternative 13: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0 77.4%

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

4. Taylor expanded in a around 0 67.0%

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

   1. unpow2 67.0%

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

   2. *-commutative 67.0%

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

   3. associate-*r* 67.0%

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

   4. unpow2 67.0%

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

   5. swap-sqr 77.4%

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

   6. unpow2 77.4%

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

   7. *-commutative 77.4%

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

   8. associate-*l* 77.4%

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

6. Simplified 77.4%

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

7. Final simplification 77.4%

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

Alternative 14: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0 77.4%

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

4. Final simplification 77.4%

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

Alternative 15: 80.3% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (/ angle_m (/ 180.0 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((angle_m / (180.0 / ((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((angle_m / (180.0 / 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((angle_m / (180.0 / 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(angle_m / Float64(180.0 / pi)))) ^ 2.0) + (b ^ 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 ^ 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[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

   1. unpow2 77.5%

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

   2. associate-/r/ 77.5%

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

   3. unpow2 77.5%

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

   4. associate-/r/ 77.5%

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

3. Simplified 77.5%

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

   1. add-sqr-sqrt 40.1%

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

   2. div-inv 40.1%

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

   3. times-frac 40.1%

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

6. Applied egg-rr 40.1%

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

7. Taylor expanded in angle around 0 77.5%

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

8. Final simplification 77.5%

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

Alternative 16: 75.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0 77.4%

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

4. Taylor expanded in angle around 0 72.5%

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

   1. *-commutative 72.5%

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

   2. associate-*l* 72.2%

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

6. Simplified 72.2%

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

   1. unpow2 72.2%

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

   2. *-commutative 72.2%

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

   3. associate-*l* 72.2%

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

   4. associate-*r* 72.3%

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

   5. associate-*r* 72.5%

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

8. Applied egg-rr 72.5%

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

9. Final simplification 72.5%

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

Alternative 17: 74.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0 77.4%

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

4. Taylor expanded in angle around 0 72.5%

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

   1. *-commutative 72.5%

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

   2. associate-*l* 72.2%

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

6. Simplified 72.2%

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

   1. unpow2 72.2%

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

   2. associate-*r* 72.2%

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

   3. associate-*r* 72.2%

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

   4. associate-*r* 72.2%

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

8. Applied egg-rr 72.2%

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

   1. *-commutative 72.2%

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

   2. associate-*r* 72.2%

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

   3. *-commutative 72.2%

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

   4. associate-*r* 72.2%

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

   5. metadata-eval 72.2%

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

   6. associate-*l* 72.2%

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

10. Simplified 72.2%

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

    1. expm1-log1p-u 71.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right) \cdot angle\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

    2. expm1-udef 70.1%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot a\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right) \cdot angle\right)\right)} - 1\right)} + {\left(b \cdot 1\right)}^{2} \]

    3. *-commutative 70.1%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\pi \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)}\right)} - 1\right) + {\left(b \cdot 1\right)}^{2} \]

    4. associate-*r* 70.1%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\pi \cdot a\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}\right)\right)\right)} - 1\right) + {\left(b \cdot 1\right)}^{2} \]

12. Applied egg-rr 70.1%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot a\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)\right)\right)} - 1\right)} + {\left(b \cdot 1\right)}^{2} \]
13. Step-by-step derivation

    1. expm1-def 71.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot a\right) \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

    2. expm1-log1p 72.2%

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

    3. associate-*l* 72.5%

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

    4. *-commutative 72.5%

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

    5. associate-*r* 72.5%

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

    6. *-commutative 72.5%

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

14. Simplified 72.5%

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

15. Final simplification 72.5%

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

Reproduce

herbie shell --seed 2024023 
(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)))