ab-angle->ABCF A

Percentage Accurate: 79.7% → 79.8%

Time: 33.1s

Alternatives: 5

Speedup: 1.3×

• 37.4% 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: 79.7% 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: 79.8% 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 82.4%

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

   1. associate-*l/ 82.4%

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

   2. associate-/l* 82.5%

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

   3. cos-neg 82.5%

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

   4. distribute-lft-neg-out 82.5%

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

   5. distribute-frac-neg 82.5%

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

   6. distribute-frac-neg 82.5%

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

   7. distribute-lft-neg-out 82.5%

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

   8. cos-neg 82.5%

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

   9. associate-*l/ 82.5%

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

   10. associate-/l* 82.5%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.9%

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

6. Final simplification 82.9%

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

Alternative 2: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. associate-*l/ 82.4%

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

   2. associate-/l* 82.5%

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

   3. cos-neg 82.5%

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

   4. distribute-lft-neg-out 82.5%

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

   5. distribute-frac-neg 82.5%

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

   6. distribute-frac-neg 82.5%

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

   7. distribute-lft-neg-out 82.5%

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

   8. cos-neg 82.5%

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

   9. associate-*l/ 82.5%

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

   10. associate-/l* 82.5%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.9%

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

6. Taylor expanded in angle around inf 82.9%

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

7. Final simplification 82.9%

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

Alternative 3: 73.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{+141}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1e+141)
   (+
    (pow b 2.0)
    (*
     (* angle PI)
     (*
      0.005555555555555556
      (* angle (* a (* a (* PI 0.005555555555555556)))))))
   (+
    (pow b 2.0)
    (*
     (* angle (* 0.005555555555555556 (* a PI)))
     (* 0.005555555555555556 (* a (* angle PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e+141) {
		tmp = pow(b, 2.0) + ((angle * ((double) M_PI)) * (0.005555555555555556 * (angle * (a * (a * (((double) M_PI) * 0.005555555555555556))))));
	} else {
		tmp = pow(b, 2.0) + ((angle * (0.005555555555555556 * (a * ((double) M_PI)))) * (0.005555555555555556 * (a * (angle * ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e+141) {
		tmp = Math.pow(b, 2.0) + ((angle * Math.PI) * (0.005555555555555556 * (angle * (a * (a * (Math.PI * 0.005555555555555556))))));
	} else {
		tmp = Math.pow(b, 2.0) + ((angle * (0.005555555555555556 * (a * Math.PI))) * (0.005555555555555556 * (a * (angle * Math.PI))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1e+141:
		tmp = math.pow(b, 2.0) + ((angle * math.pi) * (0.005555555555555556 * (angle * (a * (a * (math.pi * 0.005555555555555556))))))
	else:
		tmp = math.pow(b, 2.0) + ((angle * (0.005555555555555556 * (a * math.pi))) * (0.005555555555555556 * (a * (angle * math.pi))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1e+141)
		tmp = Float64((b ^ 2.0) + Float64(Float64(angle * pi) * Float64(0.005555555555555556 * Float64(angle * Float64(a * Float64(a * Float64(pi * 0.005555555555555556)))))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(angle * Float64(0.005555555555555556 * Float64(a * pi))) * Float64(0.005555555555555556 * Float64(a * Float64(angle * pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1e+141)
		tmp = (b ^ 2.0) + ((angle * pi) * (0.005555555555555556 * (angle * (a * (a * (pi * 0.005555555555555556))))));
	else
		tmp = (b ^ 2.0) + ((angle * (0.005555555555555556 * (a * pi))) * (0.005555555555555556 * (a * (angle * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1e+141], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(angle * Pi), $MachinePrecision] * N[(0.005555555555555556 * N[(angle * N[(a * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(angle * N[(0.005555555555555556 * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.00000000000000002e141

   1. Initial program 80.5%

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

      1. associate-*l/ 80.5%

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

      2. associate-/l* 80.6%

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

      3. cos-neg 80.6%

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

      4. distribute-lft-neg-out 80.6%

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

      5. distribute-frac-neg 80.6%

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

      6. distribute-frac-neg 80.6%

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

      7. distribute-lft-neg-out 80.6%

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

      8. cos-neg 80.6%

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

      9. associate-*l/ 80.5%

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

      10. associate-/l* 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in angle around 0 81.0%

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

   6. Taylor expanded in angle around 0 75.8%

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

      1. unpow2 75.8%

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

      2. associate-*r* 75.9%

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

      3. associate-*l* 74.6%

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

      4. *-commutative 74.6%

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

      5. *-commutative 74.6%

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

      6. associate-*r* 74.7%

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

      7. associate-*l* 74.7%

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

   8. Applied egg-rr 74.7%

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

      1. associate-*r* 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*l* 76.1%

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

      4. *-commutative 76.1%

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

      5. associate-*r* 76.0%

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

      6. *-commutative 76.0%

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

      7. associate-*l* 75.4%

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

      8. associate-*l* 75.4%

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

      9. *-commutative 75.4%

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

      10. *-commutative 75.4%

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

      11. *-commutative 75.4%

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

      12. associate-*r* 75.4%

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

      13. *-commutative 75.4%

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

      14. associate-*r* 75.4%

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

      15. *-commutative 75.4%

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

      16. *-commutative 75.4%

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

      17. *-commutative 75.4%

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

      18. associate-*l* 75.4%

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

      19. *-commutative 75.4%

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

      20. *-commutative 75.4%

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

   10. Simplified 75.4%

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

   if 1.00000000000000002e141 < a

   1. Initial program 96.5%

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

      1. associate-*l/ 96.5%

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

      2. associate-/l* 96.6%

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

      3. cos-neg 96.6%

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

      4. distribute-lft-neg-out 96.6%

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

      5. distribute-frac-neg 96.6%

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

      6. distribute-frac-neg 96.6%

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

      7. distribute-lft-neg-out 96.6%

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

      8. cos-neg 96.6%

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

      9. associate-*l/ 96.6%

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

      10. associate-/l* 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in angle around 0 96.6%

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

   6. Taylor expanded in angle around 0 96.5%

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

      1. unpow2 96.5%

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

      2. associate-*r* 96.5%

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

      3. associate-*l* 87.5%

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

      4. *-commutative 87.5%

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

      5. *-commutative 87.5%

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

      6. associate-*r* 87.6%

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

      7. associate-*l* 87.6%

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

   8. Applied egg-rr 87.6%

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

      1. associate-*r* 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*l* 96.6%

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

      4. *-commutative 96.6%

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

      5. *-commutative 96.6%

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

      6. *-commutative 96.6%

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

      7. associate-*r* 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*r* 96.6%

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

   10. Simplified 96.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{+141}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 4 \cdot 10^{-69}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 4e-69)
   (+
    (pow b 2.0)
    (*
     (* 0.005555555555555556 (* a (* angle PI)))
     (* a (* 0.005555555555555556 (* angle PI)))))
   (+
    (pow b 2.0)
    (*
     (* angle PI)
     (*
      0.005555555555555556
      (* angle (* a (* a (* PI 0.005555555555555556)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 4e-69) {
		tmp = pow(b, 2.0) + ((0.005555555555555556 * (a * (angle * ((double) M_PI)))) * (a * (0.005555555555555556 * (angle * ((double) M_PI)))));
	} else {
		tmp = pow(b, 2.0) + ((angle * ((double) M_PI)) * (0.005555555555555556 * (angle * (a * (a * (((double) M_PI) * 0.005555555555555556))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 4e-69) {
		tmp = Math.pow(b, 2.0) + ((0.005555555555555556 * (a * (angle * Math.PI))) * (a * (0.005555555555555556 * (angle * Math.PI))));
	} else {
		tmp = Math.pow(b, 2.0) + ((angle * Math.PI) * (0.005555555555555556 * (angle * (a * (a * (Math.PI * 0.005555555555555556))))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 4e-69:
		tmp = math.pow(b, 2.0) + ((0.005555555555555556 * (a * (angle * math.pi))) * (a * (0.005555555555555556 * (angle * math.pi))))
	else:
		tmp = math.pow(b, 2.0) + ((angle * math.pi) * (0.005555555555555556 * (angle * (a * (a * (math.pi * 0.005555555555555556))))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 4e-69)
		tmp = Float64((b ^ 2.0) + Float64(Float64(0.005555555555555556 * Float64(a * Float64(angle * pi))) * Float64(a * Float64(0.005555555555555556 * Float64(angle * pi)))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(angle * pi) * Float64(0.005555555555555556 * Float64(angle * Float64(a * Float64(a * Float64(pi * 0.005555555555555556)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 4e-69)
		tmp = (b ^ 2.0) + ((0.005555555555555556 * (a * (angle * pi))) * (a * (0.005555555555555556 * (angle * pi))));
	else
		tmp = (b ^ 2.0) + ((angle * pi) * (0.005555555555555556 * (angle * (a * (a * (pi * 0.005555555555555556))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 4e-69], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(0.005555555555555556 * N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(angle * Pi), $MachinePrecision] * N[(0.005555555555555556 * N[(angle * N[(a * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.9999999999999999e-69

   1. Initial program 87.3%

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

      1. associate-*l/ 87.4%

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

      2. associate-/l* 87.4%

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

      3. cos-neg 87.4%

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

      4. distribute-lft-neg-out 87.4%

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

      5. distribute-frac-neg 87.4%

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

      6. distribute-frac-neg 87.4%

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

      7. distribute-lft-neg-out 87.4%

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

      8. cos-neg 87.4%

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

      9. associate-*l/ 87.4%

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

      10. associate-/l* 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in angle around 0 87.6%

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

   6. Taylor expanded in angle around 0 84.4%

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

      1. unpow2 84.4%

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

      2. associate-*r* 84.5%

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

      3. associate-*l* 81.3%

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

      4. *-commutative 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-*r* 81.3%

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

      7. associate-*l* 81.3%

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

   8. Applied egg-rr 81.3%

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

      1. associate-*r* 84.5%

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

      2. *-commutative 84.5%

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

      3. associate-*r* 84.5%

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

      4. associate-*r* 84.5%

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

      5. associate-*r* 84.4%

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

      6. *-commutative 84.4%

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

   10. Simplified 84.4%

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

   if 3.9999999999999999e-69 < angle

   1. Initial program 71.8%

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

      1. associate-*l/ 71.7%

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

      2. associate-/l* 71.9%

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

      3. cos-neg 71.9%

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

      4. distribute-lft-neg-out 71.9%

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

      5. distribute-frac-neg 71.9%

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

      6. distribute-frac-neg 71.9%

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

      7. distribute-lft-neg-out 71.9%

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

      8. cos-neg 71.9%

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

      9. associate-*l/ 71.9%

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

      10. associate-/l* 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in angle around 0 72.8%

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

   6. Taylor expanded in angle around 0 65.2%

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

      1. unpow2 65.2%

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

      2. associate-*r* 65.2%

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

      3. associate-*l* 65.2%

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

      4. *-commutative 65.2%

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

      5. *-commutative 65.2%

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

      6. associate-*r* 65.3%

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

      7. associate-*l* 65.3%

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

   8. Applied egg-rr 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-*l* 65.3%

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

      4. *-commutative 65.3%

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

      5. associate-*r* 65.3%

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

      6. *-commutative 65.3%

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

      7. associate-*l* 69.2%

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

      8. associate-*l* 69.2%

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

      9. *-commutative 69.2%

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

      10. *-commutative 69.2%

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

      11. *-commutative 69.2%

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

      12. associate-*r* 69.2%

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

      13. *-commutative 69.2%

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

      14. associate-*r* 69.2%

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

      15. *-commutative 69.2%

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

      16. *-commutative 69.2%

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

      17. *-commutative 69.2%

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

      18. associate-*l* 69.2%

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

      19. *-commutative 69.2%

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

      20. *-commutative 69.2%

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

   10. Simplified 69.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4 \cdot 10^{-69}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

   1. associate-*l/ 82.4%

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

   2. associate-/l* 82.5%

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

   3. cos-neg 82.5%

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

   4. distribute-lft-neg-out 82.5%

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

   5. distribute-frac-neg 82.5%

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

   6. distribute-frac-neg 82.5%

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

   7. distribute-lft-neg-out 82.5%

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

   8. cos-neg 82.5%

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

   9. associate-*l/ 82.5%

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

   10. associate-/l* 82.5%

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

3. Simplified 82.5%

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

5. Taylor expanded in angle around 0 82.9%

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

6. Taylor expanded in angle around 0 78.4%

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

   1. unpow2 78.4%

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

   2. associate-*r* 78.4%

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

   3. associate-*l* 76.2%

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

   4. *-commutative 76.2%

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

   5. *-commutative 76.2%

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

   6. associate-*r* 76.2%

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

   7. associate-*l* 76.2%

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

8. Applied egg-rr 76.2%

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

   1. associate-*r* 78.4%

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

   2. *-commutative 78.4%

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

   3. associate-*r* 78.4%

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

   4. associate-*r* 78.4%

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

   5. associate-*r* 78.4%

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

   6. *-commutative 78.4%

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

10. Simplified 78.4%

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

11. Final simplification 78.4%

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

Reproduce

herbie shell --seed 2024106 
(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)))