ab-angle->ABCF C

Percentage Accurate: 79.3% → 79.3%

Time: 47.8s

Alternatives: 13

Speedup: 1.0×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* (cbrt (pow PI 3.0)) (* angle 0.005555555555555556)))) 2.0)
  (pow (* b (sin (/ (* PI 0.005555555555555556) (/ 1.0 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((cbrt(pow(((double) M_PI), 3.0)) * (angle * 0.005555555555555556)))), 2.0) + pow((b * sin(((((double) M_PI) * 0.005555555555555556) / (1.0 / angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle * 0.005555555555555556)))), 2.0) + Math.pow((b * Math.sin(((Math.PI * 0.005555555555555556) / (1.0 / angle)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(cbrt((pi ^ 3.0)) * Float64(angle * 0.005555555555555556)))) ^ 2.0) + (Float64(b * sin(Float64(Float64(pi * 0.005555555555555556) / Float64(1.0 / angle)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. expm1-log1p-u 66.2%

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

6. Applied egg-rr 66.2%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 82.4%

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

   2. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   3. div-inv 82.4%

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

   4. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   5. div-inv 82.4%

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

   6. div-inv 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\color{blue}{180 \cdot \frac{1}{angle}}}\right)\right)}^{2} \]

   7. associate-/r* 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)}\right)}^{2} \]

   8. div-inv 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\color{blue}{\pi \cdot \frac{1}{180}}}{\frac{1}{angle}}\right)\right)}^{2} \]

   9. metadata-eval 82.5%

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

8. Applied egg-rr 82.5%

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

   1. add-cbrt-cube 82.5%

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

   2. pow3 82.5%

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

10. Applied egg-rr 82.5%

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

11. Final simplification 82.5%

    \[\leadsto {\left(a \cdot \cos \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* 0.005555555555555556 (/ PI (/ 1.0 angle))))) 2.0)
  (pow (* b (sin (/ PI (/ 180.0 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((0.005555555555555556 * (((double) M_PI) / (1.0 / angle))))), 2.0) + pow((b * sin((((double) M_PI) / (180.0 / angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((0.005555555555555556 * (Math.PI / (1.0 / angle))))), 2.0) + Math.pow((b * Math.sin((Math.PI / (180.0 / angle)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((0.005555555555555556 * (math.pi / (1.0 / angle))))), 2.0) + math.pow((b * math.sin((math.pi / (180.0 / angle)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(0.005555555555555556 * Float64(pi / Float64(1.0 / angle))))) ^ 2.0) + (Float64(b * sin(Float64(pi / Float64(180.0 / angle)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((0.005555555555555556 * (pi / (1.0 / angle))))) ^ 2.0) + ((b * sin((pi / (180.0 / angle)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(Pi / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 82.4%

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

   3. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   4. un-div-inv 82.4%

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

6. Applied egg-rr 82.4%

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

   1. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 82.4%

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

   3. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   4. un-div-inv 82.4%

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

8. Applied egg-rr 82.4%

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

   1. *-un-lft-identity 82.4%

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

   2. div-inv 82.5%

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

   3. times-frac 82.5%

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

   4. metadata-eval 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\color{blue}{0.005555555555555556} \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 82.5%

    \[\leadsto {\left(a \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]

11. Final simplification 82.5%

    \[\leadsto {\left(a \cdot \cos \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (/ (* PI 0.005555555555555556) (/ 1.0 angle)))) 2.0)
  (pow (* a (cos (* PI (* angle 0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin(((((double) M_PI) * 0.005555555555555556) / (1.0 / angle)))), 2.0) + pow((a * cos((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin(((Math.PI * 0.005555555555555556) / (1.0 / angle)))), 2.0) + Math.pow((a * Math.cos((Math.PI * (angle * 0.005555555555555556)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin(((math.pi * 0.005555555555555556) / (1.0 / angle)))), 2.0) + math.pow((a * math.cos((math.pi * (angle * 0.005555555555555556)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(Float64(pi * 0.005555555555555556) / Float64(1.0 / angle)))) ^ 2.0) + (Float64(a * cos(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin(((pi * 0.005555555555555556) / (1.0 / angle)))) ^ 2.0) + ((a * cos((pi * (angle * 0.005555555555555556)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. expm1-log1p-u 66.2%

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

6. Applied egg-rr 66.2%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} \]
7. Step-by-step derivation

   1. expm1-log1p-u 82.4%

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

   2. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   3. div-inv 82.4%

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

   4. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   5. div-inv 82.4%

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

   6. div-inv 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\color{blue}{180 \cdot \frac{1}{angle}}}\right)\right)}^{2} \]

   7. associate-/r* 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)}\right)}^{2} \]

   8. div-inv 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\color{blue}{\pi \cdot \frac{1}{180}}}{\frac{1}{angle}}\right)\right)}^{2} \]

   9. metadata-eval 82.5%

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

8. Applied egg-rr 82.5%

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

9. Final simplification 82.5%

   \[\leadsto {\left(b \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 4: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in angle around inf 82.4%

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

   1. associate-*r* 82.4%

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

   2. *-commutative 82.4%

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

   3. associate-*r* 82.4%

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

7. Simplified 82.4%

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

8. Final simplification 82.4%

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

Alternative 5: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (+ (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 * 0.005555555555555556);
	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 * 0.005555555555555556);
	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 * 0.005555555555555556)
	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 * 0.005555555555555556))
	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 * 0.005555555555555556);
	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 * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

5. Final simplification 82.4%

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

Alternative 6: 79.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (* angle 0.005555555555555556)))) 2.0)
  (pow (* b (sin (/ PI (/ 180.0 angle)))) 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((((double) M_PI) / (180.0 / angle)))), 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((Math.PI / (180.0 / angle)))), 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((math.pi / (180.0 / angle)))), 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 82.4%

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

   3. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   4. un-div-inv 82.4%

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

6. Applied egg-rr 82.4%

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

7. Final simplification 82.4%

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

Alternative 7: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 82.4%

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

   3. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   4. un-div-inv 82.4%

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

6. Applied egg-rr 82.4%

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

   1. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 82.4%

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

   3. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   4. un-div-inv 82.4%

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

8. Applied egg-rr 82.4%

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

9. Final simplification 82.4%

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

Alternative 8: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {a}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (/ (* PI 0.005555555555555556) (/ 1.0 angle)))) 2.0)
  (pow a 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin(((((double) M_PI) * 0.005555555555555556) / (1.0 / angle)))), 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 * 0.005555555555555556) / (1.0 / angle)))), 2.0) + Math.pow(a, 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin(((math.pi * 0.005555555555555556) / (1.0 / angle)))), 2.0) + math.pow(a, 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(Float64(pi * 0.005555555555555556) / Float64(1.0 / angle)))) ^ 2.0) + (a ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin(((pi * 0.005555555555555556) / (1.0 / angle)))) ^ 2.0) + (a ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in angle around 0 82.0%

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

   1. add-exp-log 35.5%

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

7. Applied egg-rr 35.5%

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

   1. rem-exp-log 82.0%

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

   2. *-commutative 82.0%

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

   3. associate-*r* 82.0%

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

   4. /-rgt-identity 82.0%

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

   5. associate-/r/ 82.1%

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

9. Applied egg-rr 82.1%

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

10. Final simplification 82.1%

    \[\leadsto {\left(b \cdot \sin \left(\frac{\pi \cdot 0.005555555555555556}{\frac{1}{angle}}\right)\right)}^{2} + {a}^{2} \]
11. Add Preprocessing

Alternative 9: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in angle around 0 82.0%

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

6. Final simplification 82.0%

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

Alternative 10: 79.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

   1. metadata-eval 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 82.4%

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

   3. clear-num 82.4%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   4. un-div-inv 82.4%

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

6. Applied egg-rr 82.4%

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

7. Taylor expanded in angle around 0 82.0%

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

8. Final simplification 82.0%

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

Alternative 11: 72.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in angle around 0 82.0%

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

6. Taylor expanded in angle around 0 78.5%

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

   1. unpow2 78.5%

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

   2. associate-*r* 78.5%

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

   3. *-commutative 78.5%

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

   4. associate-*l* 77.2%

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

   5. *-commutative 77.2%

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

   6. *-commutative 77.2%

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

   7. *-commutative 77.2%

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

   8. associate-*l* 77.2%

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

   9. *-commutative 77.2%

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

8. Applied egg-rr 77.2%

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

9. Taylor expanded in b around 0 77.2%

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

    1. *-commutative 77.2%

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

    2. associate-*r* 77.2%

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

11. Simplified 77.2%

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

12. Final simplification 77.2%

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

Alternative 12: 73.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot b\right)\right)\\ {a}^{2} + t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in angle around 0 82.0%

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

6. Taylor expanded in angle around 0 78.5%

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

   1. unpow2 78.5%

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

   2. *-commutative 78.5%

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

   3. associate-*l* 78.5%

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

   4. *-commutative 78.5%

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

   5. *-commutative 78.5%

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

   6. associate-*l* 78.5%

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

   7. *-commutative 78.5%

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

8. Applied egg-rr 78.5%

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

9. Final simplification 78.5%

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

Alternative 13: 73.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in angle around 0 82.0%

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

6. Taylor expanded in angle around 0 78.5%

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

   1. unpow2 78.5%

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

   2. *-commutative 78.5%

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

   3. associate-*r* 78.5%

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

   4. *-commutative 78.5%

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

   5. associate-*l* 78.5%

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

   6. *-commutative 78.5%

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

   7. associate-*r* 78.5%

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

   8. *-commutative 78.5%

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

8. Applied egg-rr 78.5%

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

9. Final simplification 78.5%

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

Reproduce

herbie shell --seed 2024066 
(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)))