ab-angle->ABCF C

Percentage Accurate: 80.0% → 79.9%

Time: 43.5s

Alternatives: 14

Speedup: 1.5×

• 36.6% 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: 80.0% 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: 79.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

   1. associate-*r/ 79.3%

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

3. Applied egg-rr 79.3%

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

   1. clear-num 79.3%

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

   2. un-div-inv 79.3%

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

5. Applied egg-rr 79.3%

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

   1. add-cube-cbrt 79.3%

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

   2. *-un-lft-identity 79.3%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}{\color{blue}{1 \cdot \frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

   3. times-frac 79.4%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{1} \cdot \frac{\sqrt[3]{\pi}}{\frac{180}{angle}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

   4. pow2 79.4%

      \[\leadsto {\left(a \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\pi}}{\frac{180}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

7. Applied egg-rr 79.4%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{{\left(\sqrt[3]{\pi}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\pi}}{\frac{180}{angle}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

8. Final simplification 79.4%

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

Alternative 2: 79.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{180}{angle}}\\ {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{1}{{t_0}^{2}} \cdot \frac{\pi}{t_0}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (/ 180.0 angle))))
   (+
    (pow (* b (sin (/ (* PI angle) 180.0))) 2.0)
    (pow (* a (cos (* (/ 1.0 (pow t_0 2.0)) (/ PI t_0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt((180.0 / angle));
	return pow((b * sin(((((double) M_PI) * angle) / 180.0))), 2.0) + pow((a * cos(((1.0 / pow(t_0, 2.0)) * (((double) M_PI) / t_0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt((180.0 / angle));
	return Math.pow((b * Math.sin(((Math.PI * angle) / 180.0))), 2.0) + Math.pow((a * Math.cos(((1.0 / Math.pow(t_0, 2.0)) * (Math.PI / t_0)))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(Float64(180.0 / angle))
	return Float64((Float64(b * sin(Float64(Float64(pi * angle) / 180.0))) ^ 2.0) + (Float64(a * cos(Float64(Float64(1.0 / (t_0 ^ 2.0)) * Float64(pi / t_0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[(180.0 / angle), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

   1. associate-*r/ 79.3%

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

3. Applied egg-rr 79.3%

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

   1. clear-num 79.3%

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

   2. un-div-inv 79.3%

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

5. Applied egg-rr 79.3%

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

   1. *-un-lft-identity 79.3%

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

   2. add-cube-cbrt 79.2%

      \[\leadsto {\left(a \cdot \cos \left(\frac{1 \cdot \pi}{\color{blue}{\left(\sqrt[3]{\frac{180}{angle}} \cdot \sqrt[3]{\frac{180}{angle}}\right) \cdot \sqrt[3]{\frac{180}{angle}}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

   3. times-frac 79.3%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{1}{\sqrt[3]{\frac{180}{angle}} \cdot \sqrt[3]{\frac{180}{angle}}} \cdot \frac{\pi}{\sqrt[3]{\frac{180}{angle}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

   4. pow2 79.3%

      \[\leadsto {\left(a \cdot \cos \left(\frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}}} \cdot \frac{\pi}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

7. Applied egg-rr 79.3%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\frac{1}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}} \cdot \frac{\pi}{\sqrt[3]{\frac{180}{angle}}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

8. Final simplification 79.3%

   \[\leadsto {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\frac{1}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}} \cdot \frac{\pi}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} \]

Alternative 3: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Simplified 79.2%

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

4. Taylor expanded in angle around inf 79.2%

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

5. Final simplification 79.2%

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

Alternative 4: 79.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (/ (* PI angle) 180.0))) 2.0)
  (pow (* a (cos (* 0.005555555555555556 (* PI angle)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. associate-*r/ 79.3%

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

3. Applied egg-rr 79.3%

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

4. Taylor expanded in angle around inf 79.2%

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

5. Final simplification 79.2%

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

Alternative 5: 79.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* 0.005555555555555556 (* PI angle)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in angle around inf 79.2%

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

3. Final simplification 79.2%

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

Alternative 6: 80.0% 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]]

Derivation

1. Initial program 79.2%

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

2. Final simplification 79.2%

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

Alternative 7: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. associate-*r/ 79.3%

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

3. Applied egg-rr 79.3%

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

4. Final simplification 79.3%

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

Alternative 8: 80.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* b (sin (* 0.005555555555555556 (* PI angle)))) 2.0) (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in angle around 0 79.1%

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

3. Taylor expanded in b around 0 68.0%

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

   1. unpow2 68.0%

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

   2. *-commutative 68.0%

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

   3. *-commutative 68.0%

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

   4. associate-*r* 68.0%

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

   5. unpow2 68.0%

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

   6. swap-sqr 79.1%

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

   7. unpow2 79.1%

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

   8. associate-*r* 79.1%

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

   9. *-commutative 79.1%

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

   10. *-commutative 79.1%

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

5. Simplified 79.1%

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

6. Final simplification 79.1%

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

Alternative 9: 80.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (pow (* b (sin (* PI (* angle 0.005555555555555556)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in angle around 0 79.1%

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

   1. add-log-exp 68.2%

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

   2. div-inv 68.2%

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

   3. metadata-eval 68.2%

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

4. Applied egg-rr 68.2%

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

5. Taylor expanded in angle around inf 79.1%

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

   1. associate-*r* 79.1%

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

   2. *-commutative 79.1%

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

   3. *-commutative 79.1%

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

7. Simplified 79.1%

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

8. Final simplification 79.1%

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

Alternative 10: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in angle around 0 79.1%

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

3. Final simplification 79.1%

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

Alternative 11: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

   1. associate-*r/ 79.3%

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

3. Applied egg-rr 79.3%

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

4. Taylor expanded in angle around 0 79.1%

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

5. Final simplification 79.1%

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

Alternative 12: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.5e-144)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (pow (* 0.005555555555555556 (* angle (* PI b))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.5e-144) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((0.005555555555555556 * (angle * (((double) M_PI) * b))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.5e-144) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((0.005555555555555556 * (angle * (Math.PI * b))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.5e-144:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((0.005555555555555556 * (angle * (math.pi * b))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.5e-144)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + (Float64(0.005555555555555556 * Float64(angle * Float64(pi * b))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.5e-144)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + ((0.005555555555555556 * (angle * (pi * b))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.5e-144], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4999999999999998e-144

   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. Step-by-step derivation

      1. add-sqr-sqrt 36.7%

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

      2. sqrt-unprod 60.8%

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

      3. associate-*r/ 60.8%

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

      4. associate-*r/ 60.8%

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

      5. frac-times 60.8%

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

      6. metadata-eval 60.8%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

      7. metadata-eval 60.8%

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

      8. frac-times 60.8%

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

      9. associate-*l/ 60.8%

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

      10. associate-*l/ 60.8%

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

      11. sqrt-unprod 43.2%

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

      12. add-sqr-sqrt 80.0%

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

      13. add-cube-cbrt 79.8%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\frac{\pi}{-180} \cdot angle} \cdot \sqrt[3]{\frac{\pi}{-180} \cdot angle}\right) \cdot \sqrt[3]{\frac{\pi}{-180} \cdot angle}\right)}\right)}^{2} \]

      14. pow3 79.8%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\frac{\pi}{-180} \cdot angle}\right)}^{3}\right)}\right)}^{2} \]

   4. Applied egg-rr 79.8%

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

   5. Taylor expanded in angle around 0 62.5%

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

   if 4.4999999999999998e-144 < b

   1. Initial program 77.4%

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

   2. Taylor expanded in angle around 0 77.3%

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

   3. Taylor expanded in angle around 0 74.6%

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

      1. *-commutative 74.6%

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

   5. Simplified 74.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 13: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.5e-144)
   (+ (pow a 2.0) (pow (* b 0.0) 2.0))
   (+ (pow a 2.0) (pow (* angle (* b (* PI 0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.5e-144) {
		tmp = pow(a, 2.0) + pow((b * 0.0), 2.0);
	} else {
		tmp = pow(a, 2.0) + pow((angle * (b * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.5e-144) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((angle * (b * (Math.PI * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.5e-144:
		tmp = math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)
	else:
		tmp = math.pow(a, 2.0) + math.pow((angle * (b * (math.pi * 0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.5e-144)
		tmp = Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + (Float64(angle * Float64(b * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.5e-144)
		tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
	else
		tmp = (a ^ 2.0) + ((angle * (b * (pi * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.5e-144], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(angle * N[(b * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4999999999999998e-144

   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. Step-by-step derivation

      1. add-sqr-sqrt 36.7%

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

      2. sqrt-unprod 60.8%

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

      3. associate-*r/ 60.8%

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

      4. associate-*r/ 60.8%

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

      5. frac-times 60.8%

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

      6. metadata-eval 60.8%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

      7. metadata-eval 60.8%

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

      8. frac-times 60.8%

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

      9. associate-*l/ 60.8%

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

      10. associate-*l/ 60.8%

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

      11. sqrt-unprod 43.2%

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

      12. add-sqr-sqrt 80.0%

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

      13. add-cube-cbrt 79.8%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\frac{\pi}{-180} \cdot angle} \cdot \sqrt[3]{\frac{\pi}{-180} \cdot angle}\right) \cdot \sqrt[3]{\frac{\pi}{-180} \cdot angle}\right)}\right)}^{2} \]

      14. pow3 79.8%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\frac{\pi}{-180} \cdot angle}\right)}^{3}\right)}\right)}^{2} \]

   4. Applied egg-rr 79.8%

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

   5. Taylor expanded in angle around 0 62.5%

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

   if 4.4999999999999998e-144 < b

   1. Initial program 77.4%

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

   2. Taylor expanded in angle around 0 77.3%

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

   3. Taylor expanded in angle around 0 74.6%

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

      1. *-commutative 74.6%

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

   5. Simplified 74.6%

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

   6. Taylor expanded in angle around 0 74.6%

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

      1. associate-*r* 74.5%

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

      2. *-commutative 74.5%

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

      3. *-commutative 74.5%

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

      4. associate-*r* 74.6%

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

   8. Simplified 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. *-commutative 74.6%

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

      3. add-sqr-sqrt 74.6%

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

      4. pow2 74.6%

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

      5. unpow2 74.6%

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

      6. sqrt-prod 42.7%

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

      7. add-sqr-sqrt 74.6%

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

      8. *-commutative 74.6%

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

      9. *-commutative 74.6%

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

      10. associate-*l* 74.7%

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

   10. Applied egg-rr 74.7%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 14: 57.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + {\left(b \cdot 0\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (+ (pow a 2.0) (pow (* b 0.0) 2.0)))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * 0.0), 2.0);
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = (a ** 2.0d0) + ((b * 0.0d0) ** 2.0d0)
end function

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + Math.pow((b * 0.0), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + math.pow((b * 0.0), 2.0)

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + (Float64(b * 0.0) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + ((b * 0.0) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * 0.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in angle around 0 79.1%

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

   1. add-sqr-sqrt 39.1%

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

   2. sqrt-unprod 58.9%

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

   3. associate-*r/ 59.0%

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

   4. associate-*r/ 59.0%

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

   5. frac-times 58.9%

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

   6. metadata-eval 58.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   7. metadata-eval 58.9%

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

   8. frac-times 59.0%

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

   9. associate-*l/ 59.0%

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

   10. associate-*l/ 58.9%

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

   11. sqrt-unprod 40.0%

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

   12. add-sqr-sqrt 79.1%

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

   13. add-cube-cbrt 78.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\frac{\pi}{-180} \cdot angle} \cdot \sqrt[3]{\frac{\pi}{-180} \cdot angle}\right) \cdot \sqrt[3]{\frac{\pi}{-180} \cdot angle}\right)}\right)}^{2} \]

   14. pow3 78.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\frac{\pi}{-180} \cdot angle}\right)}^{3}\right)}\right)}^{2} \]

4. Applied egg-rr 78.9%

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

5. Taylor expanded in angle around 0 58.6%

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

6. Final simplification 58.6%

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

Reproduce

herbie shell --seed 2023318 
(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)))