ab-angle->ABCF C

Percentage Accurate: 80.4% → 80.4%

Time: 32.0s

Alternatives: 9

Speedup: 1.3×

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\pi \cdot {\left(\frac{1}{\sqrt[3]{\frac{180}{angle}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (pow (/ 1.0 (cbrt (/ 180.0 angle))) 3.0)))) 2.0)
  (pow (* b (sin (* PI (* angle 0.005555555555555556)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((((double) M_PI) * pow((1.0 / cbrt((180.0 / angle))), 3.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.PI * Math.pow((1.0 / Math.cbrt((180.0 / angle))), 3.0)))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle * 0.005555555555555556)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(pi * (Float64(1.0 / cbrt(Float64(180.0 / angle))) ^ 3.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[Power[N[(1.0 / N[Power[N[(180.0 / angle), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

3. Simplified 82.5%

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

   1. metadata-eval 82.5%

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

   2. div-inv 82.6%

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

   3. add-cube-cbrt 82.5%

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

   4. pow3 82.6%

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

   5. div-inv 82.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{\color{blue}{angle \cdot \frac{1}{180}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. metadata-eval 82.6%

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

6. Applied egg-rr 82.6%

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

   1. metadata-eval 82.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{angle \cdot \color{blue}{\frac{1}{180}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. div-inv 82.6%

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

   3. clear-num 82.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{\color{blue}{\frac{1}{\frac{180}{angle}}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. cbrt-div 82.7%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{180}{angle}}}\right)}}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. metadata-eval 82.7%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\frac{\color{blue}{1}}{\sqrt[3]{\frac{180}{angle}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

8. Applied egg-rr 82.7%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\color{blue}{\left(\frac{1}{\sqrt[3]{\frac{180}{angle}}}\right)}}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

9. Final simplification 82.7%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\frac{1}{\sqrt[3]{\frac{180}{angle}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 80.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle 0.005555555555555556)))) 2.0)
  (pow
   (* a (cos (* PI (pow (cbrt (* angle 0.005555555555555556)) 3.0))))
   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((((double) M_PI) * pow(cbrt((angle * 0.005555555555555556)), 3.0)))), 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((Math.PI * Math.pow(Math.cbrt((angle * 0.005555555555555556)), 3.0)))), 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(pi * (cbrt(Float64(angle * 0.005555555555555556)) ^ 3.0)))) ^ 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[(Pi * N[Power[N[Power[N[(angle * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

3. Simplified 82.5%

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

   1. metadata-eval 82.5%

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

   2. div-inv 82.6%

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

   3. add-cube-cbrt 82.5%

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

   4. pow3 82.6%

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

   5. div-inv 82.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot {\left(\sqrt[3]{\color{blue}{angle \cdot \frac{1}{180}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. metadata-eval 82.6%

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

6. Applied egg-rr 82.6%

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

7. Final simplification 82.6%

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

Alternative 3: 80.4% 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 82.6%

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

3. Simplified 82.5%

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

   1. metadata-eval 82.5%

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

   2. div-inv 82.5%

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

   3. clear-num 82.5%

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

   4. un-div-inv 82.6%

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

6. Applied egg-rr 82.6%

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

7. Final simplification 82.6%

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

Alternative 4: 80.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Taylor expanded in b around 0 73.4%

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

   1. *-commutative 73.4%

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

   2. *-commutative 73.4%

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

   3. associate-*r* 73.8%

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

   4. unpow2 73.8%

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

   5. unpow2 73.8%

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

   6. swap-sqr 82.5%

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

   7. unpow2 82.5%

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

   8. associate-*r* 82.1%

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

   9. *-commutative 82.1%

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

   10. associate-*r* 82.5%

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

8. Simplified 82.5%

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

9. Final simplification 82.5%

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

Alternative 5: 80.3% 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 82.6%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.5%

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

   1. metadata-eval 82.5%

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

   2. div-inv 82.5%

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

   3. clear-num 82.5%

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

   4. un-div-inv 82.6%

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

7. Applied egg-rr 82.6%

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

8. Final simplification 82.6%

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

Alternative 6: 74.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Taylor expanded in angle around 0 77.4%

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

   1. unpow2 77.4%

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

   2. associate-*r* 77.4%

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

   3. associate-*l* 76.7%

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

   4. *-commutative 76.7%

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

   5. *-commutative 76.7%

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

   6. *-commutative 76.7%

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

   7. associate-*r* 76.7%

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

   8. *-commutative 76.7%

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

   9. associate-*l* 76.7%

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

8. Applied egg-rr 76.7%

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

9. Taylor expanded in angle around 0 76.7%

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

    1. *-commutative 76.7%

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

    2. *-commutative 76.7%

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

    3. associate-*l* 76.7%

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

    4. associate-*l* 76.7%

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

    5. *-commutative 76.7%

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

11. Simplified 76.7%

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

12. Final simplification 76.7%

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

Alternative 7: 74.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Taylor expanded in angle around 0 77.4%

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

   1. unpow2 77.4%

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

   2. associate-*r* 77.4%

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

   3. associate-*l* 76.7%

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

   4. *-commutative 76.7%

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

   5. *-commutative 76.7%

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

   6. *-commutative 76.7%

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

   7. associate-*r* 76.7%

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

   8. *-commutative 76.7%

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

   9. associate-*l* 76.7%

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

8. Applied egg-rr 76.7%

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

9. Final simplification 76.7%

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

Alternative 8: 73.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Taylor expanded in angle around 0 77.4%

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

   1. unpow2 77.4%

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

   2. associate-*r* 77.5%

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

   3. *-commutative 77.5%

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

   4. associate-*r* 77.5%

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

   5. associate-*r* 77.5%

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

   6. *-commutative 77.5%

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

   7. associate-*r* 77.5%

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

   8. *-commutative 77.5%

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

   9. associate-*l* 77.6%

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

   10. associate-*r* 77.5%

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

   11. *-commutative 77.5%

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

8. Applied egg-rr 77.5%

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

9. Final simplification 77.5%

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

Alternative 9: 73.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.5%

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

6. Taylor expanded in angle around 0 77.4%

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

   1. unpow2 77.4%

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

   2. associate-*r* 77.4%

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

   3. associate-*r* 77.5%

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

   4. *-commutative 77.5%

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

   5. associate-*r* 77.5%

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

   6. *-commutative 77.5%

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

   7. associate-*l* 77.5%

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

   8. *-commutative 77.5%

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

   9. *-commutative 77.5%

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

8. Applied egg-rr 77.5%

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

9. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2024059 
(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)))