ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.3%

Time: 33.0s

Alternatives: 9

Speedup: 1.3×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around inf 80.7%

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

   1. *-commutative 80.7%

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

   2. metadata-eval 80.7%

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

   3. div-inv 80.6%

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

   4. associate-/l* 80.6%

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

   5. add-cube-cbrt 80.7%

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

   6. unpow2 80.7%

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

   7. *-commutative 80.7%

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

   8. associate-*l/ 80.7%

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

   9. associate-/r/ 80.6%

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

   10. div-inv 80.6%

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

   11. pow-flip 80.7%

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

   12. metadata-eval 80.7%

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

6. Applied egg-rr 80.7%

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

   1. associate-/r* 80.6%

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

   2. associate-/r/ 80.7%

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

   3. associate-/l* 80.7%

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

   4. associate-/r/ 80.7%

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

8. Simplified 80.7%

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

9. Final simplification 80.7%

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

Alternative 2: 80.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* angle (/ PI 180.0)))) 2.0)
  (pow
   (* b (pow (cbrt (cos (* 0.005555555555555556 (* angle PI)))) 3.0))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((angle * (((double) M_PI) / 180.0)))), 2.0) + pow((b * pow(cbrt(cos((0.005555555555555556 * (angle * ((double) M_PI))))), 3.0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((angle * (Math.PI / 180.0)))), 2.0) + Math.pow((b * Math.pow(Math.cbrt(Math.cos((0.005555555555555556 * (angle * Math.PI)))), 3.0)), 2.0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around inf 80.7%

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

   1. *-commutative 80.7%

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

   2. metadata-eval 80.7%

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

   3. div-inv 80.6%

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

   4. associate-/l* 80.6%

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

   5. add-cube-cbrt 80.7%

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

   6. unpow2 80.7%

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

   7. *-commutative 80.7%

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

   8. associate-*l/ 80.7%

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

   9. associate-/r/ 80.6%

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

   10. add-cube-cbrt 80.6%

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

   11. pow3 80.6%

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

6. Applied egg-rr 80.7%

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

7. Final simplification 80.7%

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

Alternative 3: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around inf 80.7%

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

5. Final simplification 80.7%

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

Alternative 4: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 80.5%

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

5. Taylor expanded in angle around 0 80.5%

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

6. Final simplification 80.5%

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

Alternative 5: 80.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 80.5%

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

5. Final simplification 80.5%

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

Alternative 6: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 80.5%

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

5. Taylor expanded in angle around inf 80.5%

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

   1. associate-*r* 80.4%

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

   2. metadata-eval 80.4%

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

   3. associate-/r/ 80.5%

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

   4. associate-*l/ 80.5%

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

   5. *-lft-identity 80.5%

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

7. Simplified 80.5%

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

8. Final simplification 80.5%

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

Alternative 7: 73.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 80.5%

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

5. Taylor expanded in angle around 0 76.0%

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

   1. *-commutative 76.0%

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

7. Simplified 76.0%

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

   1. unpow2 76.0%

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

   2. associate-*r* 76.0%

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

   3. associate-*l* 74.2%

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

   4. associate-*r* 74.2%

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

   5. *-commutative 74.2%

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

   6. *-commutative 74.2%

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

   7. *-commutative 74.2%

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

   8. associate-*l* 74.3%

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

   9. associate-*l* 74.2%

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

9. Applied egg-rr 74.2%

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

    1. associate-*l* 74.2%

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

    2. associate-*r* 74.3%

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

    3. *-commutative 74.3%

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

    4. associate-*r* 74.2%

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

    5. *-commutative 74.2%

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

    6. associate-*r* 74.3%

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

11. Simplified 74.3%

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

12. Final simplification 74.3%

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

Alternative 8: 74.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\\ {b}^{2} + t_0 \cdot t_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 (* a PI)))))
   (+ (pow b 2.0) (* t_0 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * (a * ((double) M_PI)));
	return pow(b, 2.0) + (t_0 * t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * (a * Math.PI));
	return Math.pow(b, 2.0) + (t_0 * t_0);
}

Python

def code(a, b, angle):
	t_0 = angle * (0.005555555555555556 * (a * math.pi))
	return math.pow(b, 2.0) + (t_0 * t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(0.005555555555555556 * Float64(a * pi)))
	return Float64((b ^ 2.0) + Float64(t_0 * t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = angle * (0.005555555555555556 * (a * pi));
	tmp = (b ^ 2.0) + (t_0 * t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 80.5%

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

5. Taylor expanded in angle around 0 76.0%

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

   1. *-commutative 76.0%

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

7. Simplified 76.0%

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

   1. unpow2 76.0%

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

   2. *-commutative 76.0%

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

   3. *-commutative 76.0%

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

   4. associate-*l* 76.0%

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

   5. associate-*l* 76.0%

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

   6. associate-*l* 76.0%

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

   7. associate-*l* 76.0%

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

9. Applied egg-rr 76.0%

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

10. Final simplification 76.0%

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

Alternative 9: 74.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l/ 80.6%

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

   3. associate-*r/ 80.6%

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

   4. unpow2 80.6%

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

   5. associate-*l/ 80.6%

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

   6. associate-*r/ 80.6%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 80.5%

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

5. Taylor expanded in angle around 0 76.0%

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

   1. *-commutative 76.0%

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

7. Simplified 76.0%

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

   1. unpow2 76.0%

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

   2. associate-*r* 76.0%

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

   3. *-commutative 76.0%

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

   4. associate-*l* 76.0%

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

   5. associate-*l* 76.0%

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

   6. associate-*l* 76.0%

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

9. Applied egg-rr 76.0%

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

10. Final simplification 76.0%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))