ab-angle->ABCF C

Percentage Accurate: 79.2% → 79.2%

Time: 42.0s

Alternatives: 8

Speedup: 1.0×

• 35.7% 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: 79.2% 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: 79.2% accurate, 0.7× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (/ PI (/ -180.0 angle_m)))) 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 * cos((((double) M_PI) / (-180.0 / angle_m)))), 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 * Math.cos((Math.PI / (-180.0 / angle_m)))), 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 * math.cos((math.pi / (-180.0 / angle_m)))), 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((Float64(a * cos(Float64(pi / Float64(-180.0 / angle_m)))) ^ 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 * cos((pi / (-180.0 / angle_m)))) ^ 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[N[(a * N[Cos[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(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 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 77.5%

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

   1. associate-/r/ 77.5%

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

   2. add-sqr-sqrt 36.8%

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

   3. associate-*r* 36.8%

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

   4. div-inv 36.8%

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

   5. metadata-eval 36.8%

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

6. Applied egg-rr 36.8%

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

7. Final simplification 36.8%

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

Alternative 2: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi / -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]]

Derivation

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 77.5%

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

5. Final simplification 77.5%

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

Alternative 3: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 77.5%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around inf 76.6%

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

7. Final simplification 76.6%

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

Alternative 4: 79.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 77.5%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Final simplification 76.6%

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

Alternative 5: 65.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-80}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(\left(\pi \cdot angle_m\right) \cdot \left(angle_m \cdot \left(-0.005555555555555556 \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.7e-80)
   (pow a 2.0)
   (+
    (pow a 2.0)
    (*
     b
     (*
      (* PI angle_m)
      (*
       angle_m
       (* -0.005555555555555556 (* -0.005555555555555556 (* PI b)))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.7e-80) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (b * ((((double) M_PI) * angle_m) * (angle_m * (-0.005555555555555556 * (-0.005555555555555556 * (((double) M_PI) * b))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.7e-80) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (b * ((Math.PI * angle_m) * (angle_m * (-0.005555555555555556 * (-0.005555555555555556 * (Math.PI * b))))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 2.7e-80:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (b * ((math.pi * angle_m) * (angle_m * (-0.005555555555555556 * (-0.005555555555555556 * (math.pi * b))))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.7e-80)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(b * Float64(Float64(pi * angle_m) * Float64(angle_m * Float64(-0.005555555555555556 * Float64(-0.005555555555555556 * Float64(pi * b)))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.7e-80)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (b * ((pi * angle_m) * (angle_m * (-0.005555555555555556 * (-0.005555555555555556 * (pi * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.7e-80], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(b * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(-0.005555555555555556 * N[(-0.005555555555555556 * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002e-80

   1. Initial program 74.9%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in angle around 0 74.3%

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

   6. Taylor expanded in angle around 0 65.1%

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

      1. unpow2 65.1%

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

      2. associate-*l* 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*l* 65.1%

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

      5. *-commutative 65.1%

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

      6. associate-*r* 65.1%

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

      7. *-commutative 65.1%

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

      8. associate-*l* 65.1%

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

      9. *-commutative 65.1%

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

   8. Applied egg-rr 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. rem-square-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. rem-square-sqrt 50.7%

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

      8. fabs-mul 50.7%

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

      9. unpow1 50.7%

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

      10. sqr-pow 30.3%

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

      11. fabs-sqr 30.3%

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

      12. sqr-pow 64.6%

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

      13. unpow1 64.6%

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

      14. associate-*r* 64.6%

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

      15. *-commutative 64.6%

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

      16. *-commutative 64.6%

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

      17. fabs-mul 64.6%

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

   10. Simplified 41.2%

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

   11. Taylor expanded in a around inf 56.1%

       \[\leadsto \color{blue}{{a}^{2}} \]

   if 2.7000000000000002e-80 < b

   1. Initial program 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in angle around 0 82.7%

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

   6. Taylor expanded in angle around 0 80.4%

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

      1. unpow2 80.4%

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

      2. associate-*l* 80.4%

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

      3. *-commutative 80.4%

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

      4. associate-*l* 80.4%

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

      5. *-commutative 80.4%

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

      6. associate-*r* 80.4%

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

      7. *-commutative 80.4%

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

      8. associate-*l* 80.4%

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

      9. *-commutative 80.4%

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

   8. Applied egg-rr 80.4%

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

      1. associate-*r* 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-*r* 80.4%

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

      4. associate-*l* 80.4%

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

      5. *-commutative 80.4%

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

      6. associate-*l* 80.4%

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

      7. *-commutative 80.4%

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

   10. Simplified 80.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-80}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(\left(\pi \cdot angle\right) \cdot \left(angle \cdot \left(-0.005555555555555556 \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle_m \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\\ \mathbf{if}\;b \leq 4 \cdot 10^{-80}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + t_0 \cdot t_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* -0.005555555555555556 (* PI b)))))
   (if (<= b 4e-80) (pow a 2.0) (+ (pow a 2.0) (* t_0 t_0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (-0.005555555555555556 * (((double) M_PI) * b));
	double tmp;
	if (b <= 4e-80) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 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 * (Math.PI * b));
	double tmp;
	if (b <= 4e-80) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 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 * (math.pi * b))
	tmp = 0
	if b <= 4e-80:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 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(pi * b)))
	tmp = 0.0
	if (b <= 4e-80)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 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 * (pi * b));
	tmp = 0.0;
	if (b <= 4e-80)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 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[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4e-80], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.99999999999999985e-80

   1. Initial program 74.9%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in angle around 0 74.3%

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

   6. Taylor expanded in angle around 0 65.1%

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

      1. unpow2 65.1%

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

      2. associate-*l* 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*l* 65.1%

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

      5. *-commutative 65.1%

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

      6. associate-*r* 65.1%

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

      7. *-commutative 65.1%

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

      8. associate-*l* 65.1%

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

      9. *-commutative 65.1%

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

   8. Applied egg-rr 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. rem-square-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. rem-square-sqrt 50.7%

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

      8. fabs-mul 50.7%

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

      9. unpow1 50.7%

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

      10. sqr-pow 30.3%

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

      11. fabs-sqr 30.3%

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

      12. sqr-pow 64.6%

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

      13. unpow1 64.6%

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

      14. associate-*r* 64.6%

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

      15. *-commutative 64.6%

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

      16. *-commutative 64.6%

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

      17. fabs-mul 64.6%

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

   10. Simplified 41.2%

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

   11. Taylor expanded in a around inf 56.1%

       \[\leadsto \color{blue}{{a}^{2}} \]

   if 3.99999999999999985e-80 < b

   1. Initial program 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in angle around 0 82.7%

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

   6. Taylor expanded in angle around 0 80.4%

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

      1. unpow2 80.4%

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

      2. associate-*r* 80.3%

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

      3. *-commutative 80.3%

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

      4. associate-*l* 80.4%

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

      5. *-commutative 80.4%

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

      6. associate-*r* 80.3%

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

      7. *-commutative 80.3%

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

      8. associate-*l* 80.3%

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

      9. *-commutative 80.3%

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

   8. Applied egg-rr 80.3%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-80}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(angle \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(angle \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-79}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + -0.005555555555555556 \cdot \left(\left(angle_m \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\pi \cdot angle_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.65e-79)
   (pow a 2.0)
   (+
    (pow a 2.0)
    (*
     -0.005555555555555556
     (*
      (* angle_m (* -0.005555555555555556 (* PI b)))
      (* b (* PI angle_m)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.65e-79) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (-0.005555555555555556 * ((angle_m * (-0.005555555555555556 * (((double) M_PI) * b))) * (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 <= 1.65e-79) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (-0.005555555555555556 * ((angle_m * (-0.005555555555555556 * (Math.PI * b))) * (b * (Math.PI * angle_m))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.65e-79:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (-0.005555555555555556 * ((angle_m * (-0.005555555555555556 * (math.pi * b))) * (b * (math.pi * angle_m))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.65e-79)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(-0.005555555555555556 * Float64(Float64(angle_m * Float64(-0.005555555555555556 * Float64(pi * b))) * 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 <= 1.65e-79)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (-0.005555555555555556 * ((angle_m * (-0.005555555555555556 * (pi * b))) * (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, 1.65e-79], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(-0.005555555555555556 * N[(N[(angle$95$m * N[(-0.005555555555555556 * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6499999999999999e-79

   1. Initial program 74.9%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in angle around 0 74.3%

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

   6. Taylor expanded in angle around 0 65.1%

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

      1. unpow2 65.1%

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

      2. associate-*l* 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*l* 65.1%

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

      5. *-commutative 65.1%

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

      6. associate-*r* 65.1%

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

      7. *-commutative 65.1%

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

      8. associate-*l* 65.1%

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

      9. *-commutative 65.1%

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

   8. Applied egg-rr 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*r* 64.6%

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

      3. *-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. rem-square-sqrt 36.8%

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

      6. fabs-sqr 36.8%

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

      7. rem-square-sqrt 50.7%

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

      8. fabs-mul 50.7%

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

      9. unpow1 50.7%

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

      10. sqr-pow 30.3%

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

      11. fabs-sqr 30.3%

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

      12. sqr-pow 64.6%

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

      13. unpow1 64.6%

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

      14. associate-*r* 64.6%

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

      15. *-commutative 64.6%

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

      16. *-commutative 64.6%

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

      17. fabs-mul 64.6%

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

   10. Simplified 41.2%

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

   11. Taylor expanded in a around inf 56.1%

       \[\leadsto \color{blue}{{a}^{2}} \]

   if 1.6499999999999999e-79 < b

   1. Initial program 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in angle around 0 82.7%

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

   6. Taylor expanded in angle around 0 80.4%

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

      1. unpow2 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-*r* 80.4%

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

      4. associate-*r* 80.3%

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

      5. *-commutative 80.3%

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

      6. associate-*l* 80.3%

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

      7. *-commutative 80.3%

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

      8. *-commutative 80.3%

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

      9. associate-*l* 80.4%

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

      10. *-commutative 80.4%

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

   8. Applied egg-rr 80.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-79}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + -0.005555555555555556 \cdot \left(\left(angle \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {a}^{2} \end{array} \]

FPCore

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

C

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

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = a ** 2.0d0
end function

Java

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return a ^ 2.0
end

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[Power[a, 2.0], $MachinePrecision]

Derivation

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 77.5%

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

5. Taylor expanded in angle around 0 76.6%

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

6. Taylor expanded in angle around 0 69.4%

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

   1. unpow2 69.4%

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

   2. associate-*l* 69.4%

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

   3. *-commutative 69.4%

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

   4. associate-*l* 69.4%

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

   5. *-commutative 69.4%

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

   6. associate-*r* 69.4%

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

   7. *-commutative 69.4%

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

   8. associate-*l* 69.4%

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

   9. *-commutative 69.4%

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

8. Applied egg-rr 69.4%

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

   1. *-commutative 69.4%

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

   2. associate-*r* 67.6%

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

   3. *-commutative 67.6%

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

   4. *-commutative 67.6%

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

   5. rem-square-sqrt 36.5%

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

   6. fabs-sqr 36.5%

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

   7. rem-square-sqrt 50.9%

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

   8. fabs-mul 50.9%

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

   9. unpow1 50.9%

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

   10. sqr-pow 26.2%

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

   11. fabs-sqr 26.2%

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

   12. sqr-pow 54.8%

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

   13. unpow1 54.8%

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

   14. associate-*r* 54.8%

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

   15. *-commutative 54.8%

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

   16. *-commutative 54.8%

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

   17. fabs-mul 54.8%

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

10. Simplified 38.0%

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

11. Taylor expanded in a around inf 52.7%

    \[\leadsto \color{blue}{{a}^{2}} \]

12. Final simplification 52.7%

    \[\leadsto {a}^{2} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(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)))