ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.5%

Time: 22.5s

Alternatives: 10

Speedup: N/A×

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 80.0%

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

   1. div-inv 80.0%

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

   2. metadata-eval 80.0%

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

   3. rem-cube-cbrt 80.0%

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

7. Applied egg-rr 80.0%

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

   1. rem-cube-cbrt 80.0%

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

   2. metadata-eval 80.0%

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

   3. div-inv 80.0%

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

   4. *-commutative 80.0%

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

   5. add-sqr-sqrt 42.1%

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

   6. associate-*r* 42.1%

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

   7. div-inv 42.1%

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

   8. metadata-eval 42.1%

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

9. Applied egg-rr 42.1%

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

10. Final simplification 42.1%

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

Alternative 2: 79.3% accurate, 0.8× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (pow
   (* b (sin (pow (cbrt (* (* PI -0.005555555555555556) angle_m)) 3.0)))
   2.0)))

C

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

Java

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((a ^ 2.0) + (Float64(b * sin((cbrt(Float64(Float64(pi * -0.005555555555555556) * angle_m)) ^ 3.0))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 80.0%

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

   1. div-inv 80.0%

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

   2. metadata-eval 80.0%

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

   3. rem-cube-cbrt 80.0%

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

7. Applied egg-rr 80.0%

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

8. Final simplification 80.0%

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

Alternative 3: 79.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Final simplification 80.0%

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

Alternative 4: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle_m \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq 1.75 \cdot 10^{-85}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\right)}^{2} + t_0 \cdot t_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* -0.005555555555555556 (* b PI)))))
   (if (<= b 1.75e-85)
     (pow a 2.0)
     (+ (pow (* a (cos (* angle_m (/ PI -180.0)))) 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 * (-0.005555555555555556 * (b * ((double) M_PI)));
	double tmp;
	if (b <= 1.75e-85) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow((a * cos((angle_m * (((double) M_PI) / -180.0)))), 2.0) + (t_0 * t_0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (-0.005555555555555556 * (b * Math.PI));
	double tmp;
	if (b <= 1.75e-85) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((angle_m * (Math.PI / -180.0)))), 2.0) + (t_0 * t_0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = angle_m * (-0.005555555555555556 * (b * math.pi))
	tmp = 0
	if b <= 1.75e-85:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow((a * math.cos((angle_m * (math.pi / -180.0)))), 2.0) + (t_0 * t_0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(angle_m * Float64(-0.005555555555555556 * Float64(b * pi)))
	tmp = 0.0
	if (b <= 1.75e-85)
		tmp = a ^ 2.0;
	else
		tmp = Float64((Float64(a * cos(Float64(angle_m * Float64(pi / -180.0)))) ^ 2.0) + Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = angle_m * (-0.005555555555555556 * (b * pi));
	tmp = 0.0;
	if (b <= 1.75e-85)
		tmp = a ^ 2.0;
	else
		tmp = ((a * cos((angle_m * (pi / -180.0)))) ^ 2.0) + (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(-0.005555555555555556 * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.75e-85], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[N[(a * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.74999999999999989e-85

   1. Initial program 77.0%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in angle around 0 77.9%

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

      1. div-inv 77.9%

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

      2. metadata-eval 77.9%

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

      3. rem-cube-cbrt 77.8%

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

   7. Applied egg-rr 77.8%

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

   8. Taylor expanded in angle around 0 63.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{0}}^{2} \]

   if 1.74999999999999989e-85 < b

   1. Initial program 86.0%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in angle around 0 84.1%

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

      1. associate-*r* 84.2%

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

   7. Simplified 84.2%

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

      1. unpow2 84.2%

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

      2. *-commutative 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*l* 84.2%

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

      5. associate-*l* 84.2%

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

      6. *-commutative 84.2%

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

      7. associate-*l* 84.2%

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

      8. associate-*l* 84.1%

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

      9. *-commutative 84.1%

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

   9. Applied egg-rr 84.1%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-85}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(angle \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right) \cdot \left(angle \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-85}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle_m \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(-0.005555555555555556 \cdot \left(angle_m \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right)\right) \cdot \left(angle_m \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 3.7e-85)
   (pow a 2.0)
   (+
    (pow (* a (cos (* angle_m (/ PI -180.0)))) 2.0)
    (*
     (* -0.005555555555555556 (* angle_m (* -0.005555555555555556 (* b PI))))
     (* angle_m (* b PI))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 3.7e-85) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow((a * cos((angle_m * (((double) M_PI) / -180.0)))), 2.0) + ((-0.005555555555555556 * (angle_m * (-0.005555555555555556 * (b * ((double) M_PI))))) * (angle_m * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 3.7e-85) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((angle_m * (Math.PI / -180.0)))), 2.0) + ((-0.005555555555555556 * (angle_m * (-0.005555555555555556 * (b * Math.PI)))) * (angle_m * (b * Math.PI)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 3.7e-85:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow((a * math.cos((angle_m * (math.pi / -180.0)))), 2.0) + ((-0.005555555555555556 * (angle_m * (-0.005555555555555556 * (b * math.pi)))) * (angle_m * (b * math.pi)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 3.7e-85)
		tmp = a ^ 2.0;
	else
		tmp = Float64((Float64(a * cos(Float64(angle_m * Float64(pi / -180.0)))) ^ 2.0) + Float64(Float64(-0.005555555555555556 * Float64(angle_m * Float64(-0.005555555555555556 * Float64(b * pi)))) * Float64(angle_m * Float64(b * pi))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 3.7e-85)
		tmp = a ^ 2.0;
	else
		tmp = ((a * cos((angle_m * (pi / -180.0)))) ^ 2.0) + ((-0.005555555555555556 * (angle_m * (-0.005555555555555556 * (b * pi)))) * (angle_m * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 3.7e-85], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[N[(a * N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(-0.005555555555555556 * N[(angle$95$m * N[(-0.005555555555555556 * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.69999999999999983e-85

   1. Initial program 77.0%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in angle around 0 77.9%

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

      1. div-inv 77.9%

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

      2. metadata-eval 77.9%

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

      3. rem-cube-cbrt 77.8%

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

   7. Applied egg-rr 77.8%

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

   8. Taylor expanded in angle around 0 63.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{0}}^{2} \]

   if 3.69999999999999983e-85 < b

   1. Initial program 86.0%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in angle around 0 84.1%

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

      1. associate-*r* 84.2%

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

   7. Simplified 84.2%

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

      1. unpow2 84.2%

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

      2. associate-*r* 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*l* 84.2%

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

      5. associate-*l* 84.3%

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

      6. *-commutative 84.3%

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

      7. associate-*l* 84.2%

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

      8. *-commutative 84.2%

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

   9. Applied egg-rr 84.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-85}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(-0.005555555555555556 \cdot \left(angle \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \pi\right)\right)\right)\right) \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+168}:\\ \;\;\;\;{a}^{2} + \left(-0.005555555555555556 \cdot angle_m\right) \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot angle_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(\left(\pi \cdot angle_m\right) \cdot \left(\left(\left(\pi \cdot -0.005555555555555556\right) \cdot angle_m\right) \cdot \left(b \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5e+168)
   (+
    (pow a 2.0)
    (*
     (* -0.005555555555555556 angle_m)
     (* (* b PI) (* b (* -0.005555555555555556 (* PI angle_m))))))
   (+
    (pow a 2.0)
    (*
     b
     (*
      (* PI angle_m)
      (*
       (* (* PI -0.005555555555555556) angle_m)
       (* b -0.005555555555555556)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5e+168) {
		tmp = pow(a, 2.0) + ((-0.005555555555555556 * angle_m) * ((b * ((double) M_PI)) * (b * (-0.005555555555555556 * (((double) M_PI) * angle_m)))));
	} else {
		tmp = pow(a, 2.0) + (b * ((((double) M_PI) * angle_m) * (((((double) M_PI) * -0.005555555555555556) * angle_m) * (b * -0.005555555555555556))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5e+168) {
		tmp = Math.pow(a, 2.0) + ((-0.005555555555555556 * angle_m) * ((b * Math.PI) * (b * (-0.005555555555555556 * (Math.PI * angle_m)))));
	} else {
		tmp = Math.pow(a, 2.0) + (b * ((Math.PI * angle_m) * (((Math.PI * -0.005555555555555556) * angle_m) * (b * -0.005555555555555556))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 5e+168:
		tmp = math.pow(a, 2.0) + ((-0.005555555555555556 * angle_m) * ((b * math.pi) * (b * (-0.005555555555555556 * (math.pi * angle_m)))))
	else:
		tmp = math.pow(a, 2.0) + (b * ((math.pi * angle_m) * (((math.pi * -0.005555555555555556) * angle_m) * (b * -0.005555555555555556))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5e+168)
		tmp = Float64((a ^ 2.0) + Float64(Float64(-0.005555555555555556 * angle_m) * Float64(Float64(b * pi) * Float64(b * Float64(-0.005555555555555556 * Float64(pi * angle_m))))));
	else
		tmp = Float64((a ^ 2.0) + Float64(b * Float64(Float64(pi * angle_m) * Float64(Float64(Float64(pi * -0.005555555555555556) * angle_m) * Float64(b * -0.005555555555555556)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 5e+168)
		tmp = (a ^ 2.0) + ((-0.005555555555555556 * angle_m) * ((b * pi) * (b * (-0.005555555555555556 * (pi * angle_m)))));
	else
		tmp = (a ^ 2.0) + (b * ((pi * angle_m) * (((pi * -0.005555555555555556) * angle_m) * (b * -0.005555555555555556))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5e+168], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(-0.005555555555555556 * angle$95$m), $MachinePrecision] * N[(N[(b * Pi), $MachinePrecision] * N[(b * N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(b * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(N[(Pi * -0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999967e168

   1. Initial program 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in angle around 0 78.0%

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

   6. Taylor expanded in angle around 0 72.8%

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

      1. unpow2 72.8%

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

      2. associate-*r* 72.8%

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

      3. associate-*l* 72.0%

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

      4. *-commutative 72.0%

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

      5. associate-*r* 72.0%

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

      6. *-commutative 72.0%

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

      7. associate-*r* 72.0%

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

      8. associate-*r* 72.1%

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

      9. *-commutative 72.1%

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

      10. associate-*r* 72.0%

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

   8. Applied egg-rr 72.0%

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

      1. *-commutative 72.0%

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

      2. *-commutative 72.0%

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

      3. *-commutative 72.0%

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

      4. associate-*r* 72.1%

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

      5. *-commutative 72.1%

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

   10. Simplified 72.1%

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

   if 4.99999999999999967e168 < b

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in angle around 0 99.6%

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

   6. Taylor expanded in angle around 0 99.5%

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

      1. unpow2 99.5%

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

      2. associate-*l* 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r* 99.7%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-*r* 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. associate-*l* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*l* 99.8%

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

      7. *-commutative 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+168}:\\ \;\;\;\;{a}^{2} + \left(-0.005555555555555556 \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\left(\left(\pi \cdot -0.005555555555555556\right) \cdot angle\right) \cdot \left(b \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-138}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + -0.005555555555555556 \cdot \left(\left(b \cdot \left(\left(\pi \cdot -0.005555555555555556\right) \cdot angle_m\right)\right) \cdot \left(b \cdot \left(\pi \cdot angle_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.8e-138)
   (pow a 2.0)
   (+
    (pow a 2.0)
    (*
     -0.005555555555555556
     (*
      (* b (* (* PI -0.005555555555555556) angle_m))
      (* b (* PI angle_m)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.8e-138) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (-0.005555555555555556 * ((b * ((((double) M_PI) * -0.005555555555555556) * angle_m)) * (b * (((double) M_PI) * angle_m))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.8e-138) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (-0.005555555555555556 * ((b * ((Math.PI * -0.005555555555555556) * angle_m)) * (b * (Math.PI * angle_m))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.8e-138:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (-0.005555555555555556 * ((b * ((math.pi * -0.005555555555555556) * angle_m)) * (b * (math.pi * angle_m))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.8e-138)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(-0.005555555555555556 * Float64(Float64(b * Float64(Float64(pi * -0.005555555555555556) * angle_m)) * Float64(b * Float64(pi * angle_m)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.8e-138)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (-0.005555555555555556 * ((b * ((pi * -0.005555555555555556) * angle_m)) * (b * (pi * angle_m))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.8e-138], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(-0.005555555555555556 * N[(N[(b * N[(N[(Pi * -0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(b * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000009e-138

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in angle around 0 77.7%

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

      1. div-inv 77.7%

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

      2. metadata-eval 77.7%

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

      3. rem-cube-cbrt 77.6%

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

   7. Applied egg-rr 77.6%

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

   8. Taylor expanded in angle around 0 62.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{0}}^{2} \]

   if 1.80000000000000009e-138 < b

   1. Initial program 85.5%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in angle around 0 85.2%

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

   6. Taylor expanded in angle around 0 83.0%

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

      1. unpow2 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*r* 83.0%

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

      4. associate-*r* 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-*r* 83.0%

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

      7. associate-*r* 83.1%

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

      8. *-commutative 83.1%

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

      9. associate-*r* 83.1%

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

      10. *-commutative 83.1%

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

      11. associate-*l* 83.1%

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

      12. *-commutative 83.1%

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

   8. Applied egg-rr 83.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-138}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + -0.005555555555555556 \cdot \left(\left(b \cdot \left(\left(\pi \cdot -0.005555555555555556\right) \cdot angle\right)\right) \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Taylor expanded in angle around 0 75.3%

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

   1. unpow2 75.3%

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

   2. associate-*l* 75.3%

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

   3. *-commutative 75.3%

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

   4. associate-*l* 75.4%

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

   5. *-commutative 75.4%

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

   6. associate-*r* 75.3%

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

   7. *-commutative 75.3%

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

   8. associate-*r* 75.4%

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

   9. associate-*r* 75.4%

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

   10. *-commutative 75.4%

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

   11. associate-*r* 75.4%

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

8. Applied egg-rr 75.4%

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

   1. associate-*l* 73.6%

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

   2. *-commutative 73.6%

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

   3. associate-*r* 73.6%

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

   4. *-commutative 73.6%

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

10. Simplified 73.6%

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

11. Final simplification 73.6%

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

Alternative 9: 73.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Taylor expanded in angle around 0 75.3%

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

   1. unpow2 75.3%

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

   2. associate-*r* 75.3%

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

   3. associate-*l* 73.2%

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

   4. *-commutative 73.2%

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

   5. associate-*r* 73.1%

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

   6. *-commutative 73.1%

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

   7. associate-*r* 73.2%

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

   8. associate-*r* 73.2%

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

   9. *-commutative 73.2%

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

   10. associate-*r* 73.2%

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

8. Applied egg-rr 73.2%

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

   1. associate-*r* 75.4%

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

   2. *-commutative 75.4%

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

   3. associate-*l* 73.5%

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

   4. associate-*r* 73.6%

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

   5. *-commutative 73.6%

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

   6. *-commutative 73.6%

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

   7. associate-*l* 73.6%

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

   8. *-commutative 73.6%

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

   9. *-commutative 73.6%

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

10. Simplified 73.6%

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

11. Final simplification 73.6%

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

Alternative 10: 74.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.5%

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

5. Taylor expanded in angle around 0 80.0%

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

6. Taylor expanded in angle around 0 75.3%

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

   1. unpow2 75.3%

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

   2. *-commutative 75.3%

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

   3. associate-*r* 75.3%

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

   4. associate-*r* 75.3%

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

   5. *-commutative 75.3%

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

   6. associate-*r* 75.3%

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

   7. associate-*r* 75.4%

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

   8. *-commutative 75.4%

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

   9. associate-*r* 75.3%

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

   10. *-commutative 75.3%

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

   11. associate-*l* 75.4%

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

   12. *-commutative 75.4%

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

8. Applied egg-rr 75.4%

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

9. Final simplification 75.4%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))