ab-angle->ABCF C

Percentage Accurate: 79.2% → 79.1%

Time: 52.6s

Alternatives: 11

Speedup: N/A×

• 36.2% 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: 79.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sqrt[N[(180.0 / angle$95$m), $MachinePrecision]], $MachinePrecision]}, 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[(N[(Pi / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.2%

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

   1. clear-num 80.2%

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

   2. un-div-inv 80.2%

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

3. Applied egg-rr 80.2%

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

   1. *-un-lft-identity 80.2%

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

   2. add-sqr-sqrt 40.0%

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

   3. times-frac 40.1%

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

5. Applied egg-rr 40.1%

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

   1. associate-*l/ 40.1%

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

   2. *-lft-identity 40.1%

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

7. Simplified 40.1%

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

8. Final simplification 40.1%

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

Alternative 2: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

4. Taylor expanded in angle around inf 80.1%

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

5. Final simplification 80.1%

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

Alternative 3: 79.1% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ 180.0 angle_m)))) 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) / (180.0 / angle_m)))), 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 / (180.0 / angle_m)))), 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 / (180.0 / angle_m)))), 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(180.0 / angle_m)))) ^ 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 / (180.0 / angle_m)))) ^ 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[(180.0 / angle$95$m), $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 80.2%

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

   1. clear-num 80.2%

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

   2. un-div-inv 80.2%

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

3. Applied egg-rr 80.2%

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

4. Taylor expanded in angle around inf 80.1%

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

5. Final simplification 80.1%

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

Alternative 4: 79.1% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle_m 180.0)))) 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 * cos((((double) M_PI) * (angle_m / 180.0)))), 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 * Math.cos((Math.PI * (angle_m / 180.0)))), 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 * math.cos((math.pi * (angle_m / 180.0)))), 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((Float64(a * cos(Float64(pi * Float64(angle_m / 180.0)))) ^ 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 * cos((pi * (angle_m / 180.0)))) ^ 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[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[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in angle around inf 80.2%

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

3. Final simplification 80.2%

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

Alternative 5: 79.2% 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 80.2%

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

2. Taylor expanded in angle around inf 80.2%

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

   1. associate-*r* 80.0%

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

   2. *-commutative 80.0%

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

   3. *-commutative 80.0%

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

   4. *-commutative 80.0%

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

4. Simplified 80.2%

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

5. Final simplification 80.2%

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

Alternative 6: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

Java

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

Python

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. clear-num 80.2%

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

   2. un-div-inv 80.2%

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

3. Applied egg-rr 80.2%

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

4. Final simplification 80.2%

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

Alternative 7: 79.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in angle around 0 80.0%

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

3. Taylor expanded in angle around inf 80.0%

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

4. Final simplification 80.0%

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

Alternative 8: 79.1% accurate, 1.5× 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 80.2%

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

2. Taylor expanded in angle around 0 80.0%

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

3. Taylor expanded in angle around inf 80.0%

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

   1. associate-*r* 80.0%

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

   2. *-commutative 80.0%

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

   3. *-commutative 80.0%

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

   4. *-commutative 80.0%

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

5. Simplified 80.0%

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

6. Final simplification 80.0%

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

Alternative 9: 79.1% accurate, 1.5× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 80.2%

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

   1. clear-num 80.2%

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

   2. un-div-inv 80.2%

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

3. Applied egg-rr 80.2%

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

4. Taylor expanded in angle around 0 80.1%

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

5. Final simplification 80.1%

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

Alternative 10: 74.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in angle around 0 80.0%

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

3. Taylor expanded in angle around 0 73.9%

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

   1. expm1-log1p-u 73.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)\right)} \]

   2. expm1-udef 70.9%

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

   3. *-commutative 70.9%

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

   4. associate-*r* 70.9%

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

   5. *-commutative 70.9%

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

   6. *-commutative 70.9%

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

   7. associate-*l* 70.9%

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

5. Applied egg-rr 70.9%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot b\right)\right)}^{2}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 73.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot b\right)\right)}^{2}\right)\right)} \]

   2. expm1-log1p 73.9%

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

   3. associate-*r* 73.9%

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

   4. associate-*r* 73.9%

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

   5. *-commutative 73.9%

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

   6. metadata-eval 73.9%

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

   7. associate-/r/ 73.9%

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

   8. associate-*l/ 73.9%

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

   9. associate-/r/ 73.9%

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

   10. *-commutative 73.9%

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

   11. associate-/l* 73.9%

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

   12. metadata-eval 73.9%

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

   13. associate-/r/ 73.9%

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

   14. associate-/r* 73.9%

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

   15. associate-/r/ 73.9%

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

   16. *-commutative 73.9%

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

   17. associate-/r/ 73.9%

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

7. Simplified 73.9%

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

8. Final simplification 73.9%

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

Alternative 11: 74.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in angle around 0 80.0%

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

3. Taylor expanded in angle around 0 73.9%

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

   1. expm1-log1p-u 73.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)\right)} \]

   2. expm1-udef 70.9%

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

   3. *-commutative 70.9%

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

   4. associate-*r* 70.9%

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

   5. *-commutative 70.9%

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

   6. *-commutative 70.9%

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

   7. associate-*l* 70.9%

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

5. Applied egg-rr 70.9%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot b\right)\right)}^{2}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 73.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot b\right)\right)}^{2}\right)\right)} \]

   2. expm1-log1p 73.9%

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

   3. associate-*r* 73.9%

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

   4. associate-*r* 73.9%

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

   5. *-commutative 73.9%

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

   6. metadata-eval 73.9%

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

   7. associate-/r/ 73.9%

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

   8. associate-*l/ 73.9%

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

   9. associate-/r/ 73.9%

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

   10. *-commutative 73.9%

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

   11. associate-/l* 73.9%

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

   12. metadata-eval 73.9%

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

   13. associate-/r/ 73.9%

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

   14. associate-/r* 73.9%

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

   15. associate-/r/ 73.9%

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

   16. *-commutative 73.9%

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

   17. associate-/r/ 73.9%

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

7. Simplified 73.9%

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

   1. *-commutative 73.9%

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

   2. clear-num 74.0%

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

   3. un-div-inv 74.0%

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

9. Applied egg-rr 74.0%

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

10. Final simplification 74.0%

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

Reproduce

herbie shell --seed 2023337 
(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)))