ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.2%

Time: 49.7s

Alternatives: 7

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. associate-*l/ 78.9%

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

   2. associate-/l* 78.9%

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

   3. cos-neg 78.9%

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

   4. distribute-lft-neg-out 78.9%

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

   5. distribute-frac-neg 78.9%

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

   6. distribute-frac-neg 78.9%

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

   7. distribute-lft-neg-out 78.9%

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

   8. cos-neg 78.9%

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

   9. associate-*l/ 79.0%

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

   10. associate-/l* 79.0%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around inf 79.0%

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

6. Final simplification 79.0%

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

Alternative 2: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0) + ((b * cos((angle * (pi / 180.0)))) ^ 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[N[(b * N[Cos[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

   1. associate-*l/ 78.9%

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

   2. associate-/l* 78.9%

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

   3. cos-neg 78.9%

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

   4. distribute-lft-neg-out 78.9%

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

   5. distribute-frac-neg 78.9%

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

   6. distribute-frac-neg 78.9%

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

   7. distribute-lft-neg-out 78.9%

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

   8. cos-neg 78.9%

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

   9. associate-*l/ 79.0%

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

   10. associate-/l* 79.0%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around inf 79.0%

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

6. Final simplification 79.0%

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

Alternative 3: 80.1% accurate, 1.3× 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 78.8%

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

   1. associate-*l/ 78.9%

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

   2. associate-/l* 78.9%

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

   3. cos-neg 78.9%

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

   4. distribute-lft-neg-out 78.9%

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

   5. distribute-frac-neg 78.9%

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

   6. distribute-frac-neg 78.9%

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

   7. distribute-lft-neg-out 78.9%

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

   8. cos-neg 78.9%

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

   9. associate-*l/ 79.0%

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

   10. associate-/l* 79.0%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Taylor expanded in angle around 0 78.4%

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

7. Final simplification 78.4%

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

Alternative 4: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. associate-*l/ 78.9%

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

   2. associate-/l* 78.9%

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

   3. cos-neg 78.9%

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

   4. distribute-lft-neg-out 78.9%

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

   5. distribute-frac-neg 78.9%

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

   6. distribute-frac-neg 78.9%

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

   7. distribute-lft-neg-out 78.9%

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

   8. cos-neg 78.9%

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

   9. associate-*l/ 79.0%

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

   10. associate-/l* 79.0%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Final simplification 78.4%

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

Alternative 5: 75.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{b}^{2} + \left(a \cdot 0.005555555555555556\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(angle \cdot \left(\left(a \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 4e-6)
   (+
    (pow b 2.0)
    (*
     (* a 0.005555555555555556)
     (* (* angle PI) (* angle (* a (* PI 0.005555555555555556))))))
   (+
    (pow b 2.0)
    (*
     (* angle PI)
     (*
      angle
      (* (* a 0.005555555555555556) (* PI (* a 0.005555555555555556))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 4e-6) {
		tmp = pow(b, 2.0) + ((a * 0.005555555555555556) * ((angle * ((double) M_PI)) * (angle * (a * (((double) M_PI) * 0.005555555555555556)))));
	} else {
		tmp = pow(b, 2.0) + ((angle * ((double) M_PI)) * (angle * ((a * 0.005555555555555556) * (((double) M_PI) * (a * 0.005555555555555556)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 4e-6) {
		tmp = Math.pow(b, 2.0) + ((a * 0.005555555555555556) * ((angle * Math.PI) * (angle * (a * (Math.PI * 0.005555555555555556)))));
	} else {
		tmp = Math.pow(b, 2.0) + ((angle * Math.PI) * (angle * ((a * 0.005555555555555556) * (Math.PI * (a * 0.005555555555555556)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 4e-6:
		tmp = math.pow(b, 2.0) + ((a * 0.005555555555555556) * ((angle * math.pi) * (angle * (a * (math.pi * 0.005555555555555556)))))
	else:
		tmp = math.pow(b, 2.0) + ((angle * math.pi) * (angle * ((a * 0.005555555555555556) * (math.pi * (a * 0.005555555555555556)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 4e-6)
		tmp = Float64((b ^ 2.0) + Float64(Float64(a * 0.005555555555555556) * Float64(Float64(angle * pi) * Float64(angle * Float64(a * Float64(pi * 0.005555555555555556))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(angle * pi) * Float64(angle * Float64(Float64(a * 0.005555555555555556) * Float64(pi * Float64(a * 0.005555555555555556))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 4e-6)
		tmp = (b ^ 2.0) + ((a * 0.005555555555555556) * ((angle * pi) * (angle * (a * (pi * 0.005555555555555556)))));
	else
		tmp = (b ^ 2.0) + ((angle * pi) * (angle * ((a * 0.005555555555555556) * (pi * (a * 0.005555555555555556)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 4e-6], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * 0.005555555555555556), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * N[(angle * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(angle * Pi), $MachinePrecision] * N[(angle * N[(N[(a * 0.005555555555555556), $MachinePrecision] * N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.99999999999999982e-6

   1. Initial program 84.4%

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

      1. associate-*l/ 84.5%

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

      2. associate-/l* 84.5%

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

      3. cos-neg 84.5%

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

      4. distribute-lft-neg-out 84.5%

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

      5. distribute-frac-neg 84.5%

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

      6. distribute-frac-neg 84.5%

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

      7. distribute-lft-neg-out 84.5%

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

      8. cos-neg 84.5%

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

      9. associate-*l/ 84.6%

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

      10. associate-/l* 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in angle around 0 83.8%

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

   6. Taylor expanded in angle around 0 79.8%

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

      1. unpow2 79.8%

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

      2. associate-*r* 79.9%

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

      3. associate-*l* 78.4%

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

      4. *-commutative 78.4%

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

      5. *-commutative 78.4%

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

      6. associate-*r* 78.4%

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

      7. *-commutative 78.4%

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

      8. associate-*l* 78.4%

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

   8. Applied egg-rr 78.4%

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

   if 3.99999999999999982e-6 < angle

   1. Initial program 62.8%

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

      1. associate-*l/ 63.2%

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

      2. associate-/l* 63.1%

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

      3. cos-neg 63.1%

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

      4. distribute-lft-neg-out 63.1%

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

      5. distribute-frac-neg 63.1%

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

      6. distribute-frac-neg 63.1%

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

      7. distribute-lft-neg-out 63.1%

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

      8. cos-neg 63.1%

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

      9. associate-*l/ 63.0%

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

      10. associate-/l* 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in angle around 0 63.1%

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

   6. Taylor expanded in angle around 0 51.8%

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

      1. unpow2 51.8%

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

      2. associate-*r* 51.8%

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

      3. associate-*r* 53.3%

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

      4. *-commutative 53.3%

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

      5. associate-*r* 53.3%

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

      6. *-commutative 53.3%

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

      7. associate-*l* 53.3%

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

      8. *-commutative 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. *-commutative 53.3%

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

      2. associate-*l* 57.6%

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

      3. associate-*l* 57.6%

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

   10. Simplified 57.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4 \cdot 10^{-6}:\\ \;\;\;\;{b}^{2} + \left(a \cdot 0.005555555555555556\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(angle \cdot \left(\left(a \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(angle \cdot \left(\left(a \cdot 0.005555555555555556\right) \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot t\_0\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* a 0.005555555555555556))))
   (if (<= a 4.2e-160)
     (+
      (pow b 2.0)
      (* (* angle PI) (* angle (* (* a 0.005555555555555556) t_0))))
     (+
      (pow b 2.0)
      (* (* angle t_0) (* 0.005555555555555556 (* PI (* a angle))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (a * 0.005555555555555556);
	double tmp;
	if (a <= 4.2e-160) {
		tmp = pow(b, 2.0) + ((angle * ((double) M_PI)) * (angle * ((a * 0.005555555555555556) * t_0)));
	} else {
		tmp = pow(b, 2.0) + ((angle * t_0) * (0.005555555555555556 * (((double) M_PI) * (a * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (a * 0.005555555555555556);
	double tmp;
	if (a <= 4.2e-160) {
		tmp = Math.pow(b, 2.0) + ((angle * Math.PI) * (angle * ((a * 0.005555555555555556) * t_0)));
	} else {
		tmp = Math.pow(b, 2.0) + ((angle * t_0) * (0.005555555555555556 * (Math.PI * (a * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (a * 0.005555555555555556)
	tmp = 0
	if a <= 4.2e-160:
		tmp = math.pow(b, 2.0) + ((angle * math.pi) * (angle * ((a * 0.005555555555555556) * t_0)))
	else:
		tmp = math.pow(b, 2.0) + ((angle * t_0) * (0.005555555555555556 * (math.pi * (a * angle))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(a * 0.005555555555555556))
	tmp = 0.0
	if (a <= 4.2e-160)
		tmp = Float64((b ^ 2.0) + Float64(Float64(angle * pi) * Float64(angle * Float64(Float64(a * 0.005555555555555556) * t_0))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(angle * t_0) * Float64(0.005555555555555556 * Float64(pi * Float64(a * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (a * 0.005555555555555556);
	tmp = 0.0;
	if (a <= 4.2e-160)
		tmp = (b ^ 2.0) + ((angle * pi) * (angle * ((a * 0.005555555555555556) * t_0)));
	else
		tmp = (b ^ 2.0) + ((angle * t_0) * (0.005555555555555556 * (pi * (a * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 4.2e-160], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(angle * Pi), $MachinePrecision] * N[(angle * N[(N[(a * 0.005555555555555556), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(angle * t$95$0), $MachinePrecision] * N[(0.005555555555555556 * N[(Pi * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 4.2000000000000001e-160

   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. associate-*l/ 79.2%

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

      2. associate-/l* 79.2%

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

      3. cos-neg 79.2%

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

      4. distribute-lft-neg-out 79.2%

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

      5. distribute-frac-neg 79.2%

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

      6. distribute-frac-neg 79.2%

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

      7. distribute-lft-neg-out 79.2%

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

      8. cos-neg 79.2%

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

      9. associate-*l/ 79.3%

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

      10. associate-/l* 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in angle around 0 78.4%

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

   6. Taylor expanded in angle around 0 71.6%

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

      1. unpow2 71.6%

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

      2. associate-*r* 71.6%

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

      3. associate-*r* 71.7%

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

      4. *-commutative 71.7%

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

      5. associate-*r* 71.7%

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

      6. *-commutative 71.7%

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

      7. associate-*l* 71.7%

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

      8. *-commutative 71.7%

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

   8. Applied egg-rr 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l* 70.7%

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

      3. associate-*l* 70.7%

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

   10. Simplified 70.7%

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

   if 4.2000000000000001e-160 < a

   1. Initial program 78.2%

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

      1. associate-*l/ 78.4%

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

      2. associate-/l* 78.4%

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

      3. cos-neg 78.4%

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

      4. distribute-lft-neg-out 78.4%

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

      5. distribute-frac-neg 78.4%

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

      6. distribute-frac-neg 78.4%

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

      7. distribute-lft-neg-out 78.4%

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

      8. cos-neg 78.4%

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

      9. associate-*l/ 78.4%

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

      10. associate-/l* 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in angle around 0 78.4%

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

   6. Taylor expanded in angle around 0 74.1%

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

      1. unpow2 74.1%

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

      2. associate-*r* 74.1%

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

      3. associate-*r* 69.3%

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

      4. *-commutative 69.3%

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

      5. associate-*r* 69.3%

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

      6. *-commutative 69.3%

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

      7. associate-*l* 69.3%

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

      8. *-commutative 69.3%

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

   8. Applied egg-rr 69.3%

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

      1. associate-*l* 74.1%

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

      2. associate-*l* 74.1%

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

      3. associate-*r* 74.1%

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

      4. associate-*r* 74.1%

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

   10. Simplified 74.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(angle \cdot \left(\left(a \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. associate-*l/ 78.9%

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

   2. associate-/l* 78.9%

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

   3. cos-neg 78.9%

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

   4. distribute-lft-neg-out 78.9%

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

   5. distribute-frac-neg 78.9%

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

   6. distribute-frac-neg 78.9%

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

   7. distribute-lft-neg-out 78.9%

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

   8. cos-neg 78.9%

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

   9. associate-*l/ 79.0%

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

   10. associate-/l* 79.0%

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

3. Simplified 79.0%

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

5. Taylor expanded in angle around 0 78.4%

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

6. Taylor expanded in angle around 0 72.5%

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

   1. unpow2 72.5%

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

   2. associate-*r* 72.5%

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

   3. associate-*l* 71.4%

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

   4. *-commutative 71.4%

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

   5. *-commutative 71.4%

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

   6. associate-*r* 71.5%

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

   7. *-commutative 71.5%

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

   8. associate-*l* 71.5%

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

8. Applied egg-rr 71.5%

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

9. Final simplification 71.5%

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

Reproduce

herbie shell --seed 2024076 
(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)))