ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.9%

Time: 30.8s

Alternatives: 12

Speedup: 1.5×

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. add-log-exp 79.1%

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

   2. div-inv 79.2%

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

   3. associate-*l* 79.2%

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

   4. metadata-eval 79.2%

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

3. Applied egg-rr 79.2%

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

4. Final simplification 79.2%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around inf 79.1%

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

3. Final simplification 79.1%

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

Alternative 3: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around inf 79.1%

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

   1. metadata-eval 78.9%

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

   2. associate-/r/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. *-commutative 78.9%

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

   5. associate-/l* 78.9%

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

   6. metadata-eval 78.9%

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

   7. associate-*r/ 79.0%

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

4. Simplified 79.1%

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

5. Final simplification 79.1%

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

Alternative 4: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. add-cube-cbrt 79.1%

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

   2. pow3 79.1%

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

   3. pow-to-exp 41.8%

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

   4. div-inv 41.8%

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

   5. associate-*l* 41.8%

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

   6. metadata-eval 41.8%

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

3. Applied egg-rr 41.8%

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

   1. exp-to-pow 79.1%

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

   2. pow3 79.1%

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

   3. add-cube-cbrt 79.2%

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

   4. *-commutative 79.2%

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

   5. associate-*r* 79.0%

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

   6. *-commutative 79.0%

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

   7. associate-*r* 79.2%

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

5. Applied egg-rr 79.2%

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

6. Final simplification 79.2%

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

Alternative 5: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.0%

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

3. Taylor expanded in angle around inf 78.9%

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

4. Final simplification 78.9%

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

Alternative 6: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.0%

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

3. Taylor expanded in angle around inf 78.9%

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

   1. metadata-eval 78.9%

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

   2. associate-/r/ 78.9%

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

   3. associate-/l* 78.9%

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

   4. *-commutative 78.9%

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

   5. associate-/l* 78.9%

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

   6. metadata-eval 78.9%

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

   7. associate-*r/ 79.0%

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

5. Simplified 79.0%

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

6. Final simplification 79.0%

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

Alternative 7: 79.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.0%

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

3. Final simplification 79.0%

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

Alternative 8: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 100000000:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\pi}^{2} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 100000000.0)
   (+ (pow b 2.0) (pow (* a (* angle (/ PI 180.0))) 2.0))
   (+
    (pow b 2.0)
    (* (pow PI 2.0) (* 3.08641975308642e-5 (* angle (* angle (* a a))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 100000000.0) {
		tmp = pow(b, 2.0) + pow((a * (angle * (((double) M_PI) / 180.0))), 2.0);
	} else {
		tmp = pow(b, 2.0) + (pow(((double) M_PI), 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 100000000.0) {
		tmp = Math.pow(b, 2.0) + Math.pow((a * (angle * (Math.PI / 180.0))), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (Math.pow(Math.PI, 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= 100000000.0:
		tmp = math.pow(b, 2.0) + math.pow((a * (angle * (math.pi / 180.0))), 2.0)
	else:
		tmp = math.pow(b, 2.0) + (math.pow(math.pi, 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 100000000.0)
		tmp = Float64((b ^ 2.0) + (Float64(a * Float64(angle * Float64(pi / 180.0))) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + Float64((pi ^ 2.0) * Float64(3.08641975308642e-5 * Float64(angle * Float64(angle * Float64(a * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 100000000.0)
		tmp = (b ^ 2.0) + ((a * (angle * (pi / 180.0))) ^ 2.0);
	else
		tmp = (b ^ 2.0) + ((pi ^ 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 100000000.0], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[Pi, 2.0], $MachinePrecision] * N[(3.08641975308642e-5 * N[(angle * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 1e8

   1. Initial program 87.1%

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

   2. Taylor expanded in angle around 0 86.8%

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

   3. Taylor expanded in angle around 0 82.6%

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

      1. metadata-eval 82.6%

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

      2. associate-/r/ 82.6%

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

      3. associate-/l* 82.6%

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

      4. *-commutative 82.6%

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

      5. associate-/l* 82.6%

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

      6. metadata-eval 82.6%

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

      7. associate-*r/ 82.7%

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

   5. Simplified 82.7%

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

   if 1e8 < (/.f64 angle 180)

   1. Initial program 59.5%

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

   2. Taylor expanded in angle around 0 59.9%

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

   3. Taylor expanded in angle around 0 43.1%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   4. Step-by-step derivation

      1. associate-*r* 43.1%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {a}^{2}\right) \cdot {\pi}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. associate-*r* 43.1%

         \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {a}^{2}\right)\right) \cdot {\pi}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 43.1%

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {a}^{2}\right)\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]

      4. associate-*l* 56.2%

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(angle \cdot \left(angle \cdot {a}^{2}\right)\right)}\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 56.2%

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right) \cdot {\pi}^{2}} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 100000000:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\pi}^{2} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;{b}^{2} + \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\pi}^{2} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e-66)
   (+
    (pow b 2.0)
    (* (* (* a angle) (* a angle)) (* 3.08641975308642e-5 (pow PI 2.0))))
   (+
    (pow b 2.0)
    (* (pow PI 2.0) (* 3.08641975308642e-5 (* angle (* angle (* a a))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e-66) {
		tmp = pow(b, 2.0) + (((a * angle) * (a * angle)) * (3.08641975308642e-5 * pow(((double) M_PI), 2.0)));
	} else {
		tmp = pow(b, 2.0) + (pow(((double) M_PI), 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e-66) {
		tmp = Math.pow(b, 2.0) + (((a * angle) * (a * angle)) * (3.08641975308642e-5 * Math.pow(Math.PI, 2.0)));
	} else {
		tmp = Math.pow(b, 2.0) + (Math.pow(Math.PI, 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= 2e-66:
		tmp = math.pow(b, 2.0) + (((a * angle) * (a * angle)) * (3.08641975308642e-5 * math.pow(math.pi, 2.0)))
	else:
		tmp = math.pow(b, 2.0) + (math.pow(math.pi, 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e-66)
		tmp = Float64((b ^ 2.0) + Float64(Float64(Float64(a * angle) * Float64(a * angle)) * Float64(3.08641975308642e-5 * (pi ^ 2.0))));
	else
		tmp = Float64((b ^ 2.0) + Float64((pi ^ 2.0) * Float64(3.08641975308642e-5 * Float64(angle * Float64(angle * Float64(a * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e-66)
		tmp = (b ^ 2.0) + (((a * angle) * (a * angle)) * (3.08641975308642e-5 * (pi ^ 2.0)));
	else
		tmp = (b ^ 2.0) + ((pi ^ 2.0) * (3.08641975308642e-5 * (angle * (angle * (a * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-66], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(N[(a * angle), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[Power[Pi, 2.0], $MachinePrecision] * N[(3.08641975308642e-5 * N[(angle * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 2e-66

   1. Initial program 86.7%

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

   2. Taylor expanded in angle around 0 86.5%

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

   3. Taylor expanded in angle around 0 83.8%

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

      1. *-commutative 83.8%

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

      2. metadata-eval 83.8%

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

      3. div-inv 83.8%

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

      4. associate-/l* 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. associate-/r/ 83.8%

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

   7. Simplified 83.8%

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

      1. associate-*l/ 83.8%

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

      2. associate-*r/ 83.8%

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

      3. unpow2 83.8%

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

      4. associate-*r* 83.9%

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

      5. associate-*r* 83.8%

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

      6. swap-sqr 83.8%

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

      7. *-commutative 83.8%

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

      8. *-commutative 83.8%

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

      9. div-inv 83.8%

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

      10. metadata-eval 83.8%

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

      11. *-commutative 83.8%

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

      12. div-inv 83.8%

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

      13. metadata-eval 83.8%

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

      14. *-commutative 83.8%

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

      15. swap-sqr 83.8%

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

      16. metadata-eval 83.8%

          \[\leadsto \left(\left(angle \cdot a\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot \left(\pi \cdot \pi\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      17. pow2 83.8%

          \[\leadsto \left(\left(angle \cdot a\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\pi}^{2}}\right) + {\left(b \cdot 1\right)}^{2} \]

   9. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\left(\left(angle \cdot a\right) \cdot \left(angle \cdot a\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

   if 2e-66 < (/.f64 angle 180)

   1. Initial program 65.9%

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

   2. Taylor expanded in angle around 0 65.8%

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

   3. Taylor expanded in angle around 0 49.2%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   4. Step-by-step derivation

      1. associate-*r* 49.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {a}^{2}\right) \cdot {\pi}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. associate-*r* 49.2%

         \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {a}^{2}\right)\right) \cdot {\pi}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 49.2%

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {a}^{2}\right)\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]

      4. associate-*l* 59.6%

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(angle \cdot \left(angle \cdot {a}^{2}\right)\right)}\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 59.6%

         \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]

   5. Simplified 59.6%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right) \cdot {\pi}^{2}} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-66}:\\ \;\;\;\;{b}^{2} + \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\pi}^{2} \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.0%

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

3. Taylor expanded in angle around 0 72.9%

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

   1. *-commutative 72.9%

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

   2. metadata-eval 72.9%

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

   3. div-inv 72.9%

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

   4. associate-/l* 72.9%

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

5. Applied egg-rr 72.9%

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

   1. associate-/r/ 72.9%

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

7. Simplified 72.9%

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

8. Taylor expanded in a around 0 72.9%

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

   1. *-commutative 72.9%

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

10. Simplified 72.9%

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

11. Final simplification 72.9%

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

Alternative 11: 74.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.0%

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

3. Taylor expanded in angle around 0 72.9%

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

4. Final simplification 72.9%

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

Alternative 12: 74.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0 79.0%

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

3. Taylor expanded in angle around 0 72.9%

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

   1. metadata-eval 72.9%

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

   2. associate-/r/ 72.9%

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

   3. associate-/l* 72.9%

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

   4. *-commutative 72.9%

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

   5. associate-/l* 72.9%

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

   6. metadata-eval 72.9%

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

   7. associate-*r/ 72.9%

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

5. Simplified 72.9%

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

6. Final simplification 72.9%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))