ab-angle->ABCF C

Percentage Accurate: 80.1% → 80.0%

Time: 31.8s

Alternatives: 11

Speedup: 1.0×

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.1% 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: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left({\left(\sqrt{\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)}\right)}^{2}\right)\right)}^{2} + {\left(b \cdot \sin \left(\sqrt[3]{{\pi}^{3}} \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (* a (cos (pow (sqrt (* PI (* angle_m 0.005555555555555556))) 2.0)))
   2.0)
  (pow (* b (sin (* (cbrt (pow PI 3.0)) (/ angle_m 180.0)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos(pow(sqrt((((double) M_PI) * (angle_m * 0.005555555555555556))), 2.0))), 2.0) + pow((b * sin((cbrt(pow(((double) M_PI), 3.0)) * (angle_m / 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 * Math.cos(Math.pow(Math.sqrt((Math.PI * (angle_m * 0.005555555555555556))), 2.0))), 2.0) + Math.pow((b * Math.sin((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle_m / 180.0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos((sqrt(Float64(pi * Float64(angle_m * 0.005555555555555556))) ^ 2.0))) ^ 2.0) + (Float64(b * sin(Float64(cbrt((pi ^ 3.0)) * Float64(angle_m / 180.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[Power[N[Sqrt[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

   1. clear-num 82.1%

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

   2. un-div-inv 82.2%

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

4. Applied egg-rr 82.2%

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

   1. clear-num 82.2%

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

   2. associate-/r* 82.1%

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

   3. add-sqr-sqrt 41.5%

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

   4. pow2 41.5%

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

   5. pow1 41.5%

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

   6. pow1 41.5%

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

   7. clear-num 41.1%

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

   8. *-commutative 41.1%

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

   9. associate-/l* 41.1%

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

   10. div-inv 41.1%

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

   11. metadata-eval 41.1%

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

6. Applied egg-rr 41.1%

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

   1. add-cbrt-cube 41.2%

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

   2. pow3 41.2%

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

8. Applied egg-rr 41.2%

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

9. Final simplification 41.2%

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

Alternative 2: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in angle around inf 82.1%

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

4. Final simplification 82.1%

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

Alternative 3: 80.1% 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 82.1%

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

3. Simplified 82.1%

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

5. Final simplification 82.1%

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

Alternative 4: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in angle around inf 82.1%

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

   1. *-commutative 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-*l* 82.1%

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

5. Simplified 82.1%

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

6. Final simplification 82.1%

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

Alternative 5: 80.1% accurate, 1.0× speedup

Math

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

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 (* PI (/ angle_m 180.0)))) 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((((double) M_PI) * (angle_m / 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 * Math.cos((Math.PI / (180.0 / angle_m)))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle_m / 180.0)))), 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.pi * (angle_m / 180.0)))), 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(pi * Float64(angle_m / 180.0)))) ^ 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((pi * (angle_m / 180.0)))) ^ 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[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

   1. clear-num 82.1%

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

   2. un-div-inv 82.2%

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

4. Applied egg-rr 82.2%

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

5. Final simplification 82.2%

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

Alternative 6: 79.9% 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 82.1%

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

3. Simplified 82.1%

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

5. Taylor expanded in angle around 0 77.3%

   \[\leadsto {\left(a \cdot \color{blue}{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \]

6. Taylor expanded in angle around 0 81.5%

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

7. Final simplification 81.5%

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

Alternative 7: 79.9% accurate, 1.3× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 82.1%

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

   1. clear-num 82.1%

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

   2. un-div-inv 82.2%

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

4. Applied egg-rr 82.2%

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

   1. clear-num 82.2%

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

   2. associate-/r* 82.1%

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

   3. add-sqr-sqrt 41.5%

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

   4. pow2 41.5%

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

   5. pow1 41.5%

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

   6. pow1 41.5%

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

   7. clear-num 41.1%

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

   8. *-commutative 41.1%

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

   9. associate-/l* 41.1%

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

   10. div-inv 41.1%

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

   11. metadata-eval 41.1%

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

6. Applied egg-rr 41.1%

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

7. Taylor expanded in angle around 0 81.6%

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

8. Final simplification 81.6%

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

Alternative 8: 79.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 82.1%

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

5. Taylor expanded in angle around 0 81.5%

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

6. Taylor expanded in angle around 0 81.5%

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

   1. associate-*r* 81.6%

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

   2. *-commutative 81.6%

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

   3. *-commutative 81.6%

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

8. Simplified 81.6%

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

9. Final simplification 81.6%

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

Alternative 9: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-52}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle\_m \cdot \left(\pi \cdot b\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4e-52)
   (pow a 2.0)
   (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* angle_m (* PI b)) 2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 4e-52) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (3.08641975308642e-5 * pow((angle_m * (((double) M_PI) * b)), 2.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 4e-52) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((angle_m * (Math.PI * b)), 2.0));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 4e-52:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((angle_m * (math.pi * b)), 2.0))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 4e-52)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(angle_m * Float64(pi * b)) ^ 2.0)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 4e-52)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((angle_m * (pi * b)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 4e-52], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4e-52

   1. Initial program 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in angle around 0 78.4%

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

   6. Taylor expanded in angle around 0 73.7%

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

      1. unpow2 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*l* 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-*l* 73.7%

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

      6. add-sqr-sqrt 45.8%

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

      7. sqrt-prod 60.1%

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

      8. unpow2 60.1%

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

      9. unpow-prod-down 60.1%

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

      10. sqrt-prod 60.1%

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

      11. metadata-eval 60.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(b \cdot \left(\pi \cdot angle\right)\right) \cdot \left(-0.005555555555555556 \cdot \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt{{\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right)\right) \]

      12. metadata-eval 60.1%

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

      13. pow2 60.1%

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

      14. sqrt-unprod 31.6%

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

      15. add-sqr-sqrt 50.9%

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

      16. *-commutative 50.9%

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

      17. associate-*l* 50.9%

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

   8. Applied egg-rr 50.9%

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

      1. associate-*l* 50.3%

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

      2. *-commutative 50.3%

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

      3. associate-*r* 50.3%

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

      4. metadata-eval 50.3%

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

      5. *-commutative 50.3%

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

   10. Simplified 50.3%

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

   11. Taylor expanded in a around inf 60.2%

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

   if 4e-52 < b

   1. Initial program 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in angle around 0 88.8%

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

   6. Taylor expanded in angle around 0 86.4%

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

   7. Taylor expanded in angle around 0 71.2%

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

      1. *-commutative 71.2%

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

      2. *-commutative 71.2%

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

      3. unpow2 71.2%

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

      4. unpow2 71.2%

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

      5. swap-sqr 71.3%

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

      6. unpow2 71.3%

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

      7. swap-sqr 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-*r* 86.4%

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

      10. *-commutative 86.4%

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

      11. associate-*r* 86.4%

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

      12. unpow2 86.4%

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

      13. associate-*r* 86.4%

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

      14. *-commutative 86.4%

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

      15. *-commutative 86.4%

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

   9. Simplified 86.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-52}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\_m\right)\right)\\ \mathbf{if}\;b \leq 7.2 \cdot 10^{-53}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* b (* PI angle_m)))))
   (if (<= b 7.2e-53) (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 = 0.005555555555555556 * (b * (((double) M_PI) * angle_m));
	double tmp;
	if (b <= 7.2e-53) {
		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 = 0.005555555555555556 * (b * (Math.PI * angle_m));
	double tmp;
	if (b <= 7.2e-53) {
		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 = 0.005555555555555556 * (b * (math.pi * angle_m))
	tmp = 0
	if b <= 7.2e-53:
		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(0.005555555555555556 * Float64(b * Float64(pi * angle_m)))
	tmp = 0.0
	if (b <= 7.2e-53)
		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 = 0.005555555555555556 * (b * (pi * angle_m));
	tmp = 0.0;
	if (b <= 7.2e-53)
		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[(0.005555555555555556 * N[(b * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.2e-53], 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 < 7.1999999999999998e-53

   1. Initial program 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in angle around 0 78.4%

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

   6. Taylor expanded in angle around 0 73.7%

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

      1. unpow2 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*l* 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-*l* 73.7%

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

      6. add-sqr-sqrt 45.8%

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

      7. sqrt-prod 60.1%

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

      8. unpow2 60.1%

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

      9. unpow-prod-down 60.1%

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

      10. sqrt-prod 60.1%

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

      11. metadata-eval 60.1%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(b \cdot \left(\pi \cdot angle\right)\right) \cdot \left(-0.005555555555555556 \cdot \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt{{\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right)\right) \]

      12. metadata-eval 60.1%

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

      13. pow2 60.1%

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

      14. sqrt-unprod 31.6%

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

      15. add-sqr-sqrt 50.9%

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

      16. *-commutative 50.9%

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

      17. associate-*l* 50.9%

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

   8. Applied egg-rr 50.9%

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

      1. associate-*l* 50.3%

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

      2. *-commutative 50.3%

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

      3. associate-*r* 50.3%

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

      4. metadata-eval 50.3%

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

      5. *-commutative 50.3%

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

   10. Simplified 50.3%

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

   11. Taylor expanded in a around inf 60.2%

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

   if 7.1999999999999998e-53 < b

   1. Initial program 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in angle around 0 88.8%

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

   6. Taylor expanded in angle around 0 86.4%

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

      1. unpow-prod-down 86.4%

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

      2. add-sqr-sqrt 86.4%

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

      3. sqrt-prod 86.4%

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

      4. metadata-eval 86.4%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt{{\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \cdot \sqrt{{-0.005555555555555556}^{2} \cdot {\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}} \]

      5. metadata-eval 86.4%

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

      6. pow2 86.4%

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

      7. sqrt-unprod 50.7%

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

      8. add-sqr-sqrt 69.0%

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

      9. *-commutative 69.0%

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

      10. associate-*l* 69.0%

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

      11. sqrt-prod 69.0%

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

      12. metadata-eval 69.0%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt{{\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right) \]

      13. metadata-eval 69.0%

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

      14. pow2 69.0%

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

      15. sqrt-unprod 50.8%

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

      16. add-sqr-sqrt 86.4%

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

      17. *-commutative 86.4%

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

      18. associate-*l* 86.4%

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

   8. Applied egg-rr 86.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-53}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.2% 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 82.1%

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

3. Simplified 82.1%

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

5. Taylor expanded in angle around 0 81.5%

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

6. Taylor expanded in angle around 0 77.5%

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

   1. unpow2 77.5%

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

   2. *-commutative 77.5%

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

   3. associate-*l* 77.5%

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

   4. *-commutative 77.5%

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

   5. associate-*l* 77.6%

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

   6. add-sqr-sqrt 42.7%

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

   7. sqrt-prod 57.9%

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

   8. unpow2 57.9%

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

   9. unpow-prod-down 57.9%

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

   10. sqrt-prod 57.9%

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

   11. metadata-eval 57.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(b \cdot \left(\pi \cdot angle\right)\right) \cdot \left(-0.005555555555555556 \cdot \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}}} \cdot \sqrt{{\left(angle \cdot \left(b \cdot \pi\right)\right)}^{2}}\right)\right) \]

   12. metadata-eval 57.9%

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

   13. pow2 57.9%

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

   14. sqrt-unprod 27.2%

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

   15. add-sqr-sqrt 46.2%

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

   16. *-commutative 46.2%

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

   17. associate-*l* 46.2%

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

8. Applied egg-rr 46.2%

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

   1. associate-*l* 45.8%

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

   2. *-commutative 45.8%

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

   3. associate-*r* 45.8%

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

   4. metadata-eval 45.8%

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

   5. *-commutative 45.8%

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

10. Simplified 45.8%

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

11. Taylor expanded in a around inf 58.1%

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

12. Final simplification 58.1%

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

Reproduce

herbie shell --seed 2024039 
(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)))