ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.4%

Time: 23.8s

Alternatives: 7

Speedup: N/A×

• 36.7% 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: 79.6% 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: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot \pi\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* (expm1 (log1p (* 0.005555555555555556 angle_m))) PI))) 2.0)
  (pow b 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin((expm1(log1p((0.005555555555555556 * angle_m))) * ((double) M_PI)))), 2.0) + pow(b, 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin((Math.expm1(Math.log1p((0.005555555555555556 * angle_m))) * Math.PI))), 2.0) + Math.pow(b, 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin((math.expm1(math.log1p((0.005555555555555556 * angle_m))) * math.pi))), 2.0) + math.pow(b, 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(expm1(log1p(Float64(0.005555555555555556 * angle_m))) * pi))) ^ 2.0) + (b ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(Exp[N[Log[1 + N[(0.005555555555555556 * angle$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. unpow2 82.9%

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

   2. swap-sqr 82.9%

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

   3. associate-*l* 82.9%

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

   4. associate-*l* 82.9%

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

   5. cancel-sign-sub 82.9%

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

   6. cancel-sign-sub-inv 82.9%

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

   7. associate-*l/ 82.6%

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

   8. associate-/l* 83.0%

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

3. Simplified 82.9%

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

5. Taylor expanded in angle around 0 83.0%

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

6. Taylor expanded in angle around 0 82.6%

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

   1. associate-*r* 83.0%

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

8. Simplified 83.0%

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

   1. expm1-log1p-u 64.0%

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.005555555555555556 \cdot angle\right)\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   2. expm1-undefine 56.7%

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot angle\right)} - 1\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

10. Applied egg-rr 56.7%

    \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot angle\right)} - 1\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
11. Step-by-step derivation

    1. expm1-define 64.0%

       \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.005555555555555556 \cdot angle\right)\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

12. Simplified 64.0%

    \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.005555555555555556 \cdot angle\right)\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

13. Final simplification 64.0%

    \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.005555555555555556 \cdot angle\right)\right) \cdot \pi\right)\right)}^{2} + {b}^{2} \]
14. Add Preprocessing

Alternative 2: 79.3% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow b 2.0) (pow (* a (sin (* 0.005555555555555556 (* angle_m PI)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. unpow2 82.9%

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

   2. swap-sqr 82.9%

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

   3. associate-*l* 82.9%

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

   4. associate-*l* 82.9%

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

   5. cancel-sign-sub 82.9%

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

   6. cancel-sign-sub-inv 82.9%

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

   7. associate-*l/ 82.6%

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

   8. associate-/l* 83.0%

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

3. Simplified 82.9%

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

5. Taylor expanded in angle around 0 83.0%

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

6. Taylor expanded in angle around 0 82.6%

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

7. Final simplification 82.6%

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

Alternative 3: 79.4% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow b 2.0) (pow (* a (sin (* angle_m (/ PI 180.0)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. unpow2 82.9%

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

   2. swap-sqr 82.9%

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

   3. associate-*l* 82.9%

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

   4. associate-*l* 82.9%

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

   5. cancel-sign-sub 82.9%

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

   6. cancel-sign-sub-inv 82.9%

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

   7. associate-*l/ 82.6%

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

   8. associate-/l* 83.0%

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

3. Simplified 82.9%

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

5. Taylor expanded in angle around 0 83.0%

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

6. Final simplification 83.0%

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

Alternative 4: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \left(a \cdot \pi\right)\\ \mathbf{if}\;angle\_m \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(t\_0 \cdot \left(0.005555555555555556 \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(angle\_m \cdot \pi\right) \cdot {a}^{2}\right) \cdot \frac{0.005555555555555556}{\frac{180}{angle\_m \cdot \pi}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* a PI))))
   (if (<= angle_m 1.5e-77)
     (+
      (pow b 2.0)
      (* 0.005555555555555556 (* t_0 (* 0.005555555555555556 t_0))))
     (+
      (pow b 2.0)
      (*
       (* (* angle_m PI) (pow a 2.0))
       (/ 0.005555555555555556 (/ 180.0 (* angle_m PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (a * ((double) M_PI));
	double tmp;
	if (angle_m <= 1.5e-77) {
		tmp = pow(b, 2.0) + (0.005555555555555556 * (t_0 * (0.005555555555555556 * t_0)));
	} else {
		tmp = pow(b, 2.0) + (((angle_m * ((double) M_PI)) * pow(a, 2.0)) * (0.005555555555555556 / (180.0 / (angle_m * ((double) M_PI)))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (a * Math.PI);
	double tmp;
	if (angle_m <= 1.5e-77) {
		tmp = Math.pow(b, 2.0) + (0.005555555555555556 * (t_0 * (0.005555555555555556 * t_0)));
	} else {
		tmp = Math.pow(b, 2.0) + (((angle_m * Math.PI) * Math.pow(a, 2.0)) * (0.005555555555555556 / (180.0 / (angle_m * Math.PI))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = angle_m * (a * math.pi)
	tmp = 0
	if angle_m <= 1.5e-77:
		tmp = math.pow(b, 2.0) + (0.005555555555555556 * (t_0 * (0.005555555555555556 * t_0)))
	else:
		tmp = math.pow(b, 2.0) + (((angle_m * math.pi) * math.pow(a, 2.0)) * (0.005555555555555556 / (180.0 / (angle_m * math.pi))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(angle_m * Float64(a * pi))
	tmp = 0.0
	if (angle_m <= 1.5e-77)
		tmp = Float64((b ^ 2.0) + Float64(0.005555555555555556 * Float64(t_0 * Float64(0.005555555555555556 * t_0))));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(Float64(angle_m * pi) * (a ^ 2.0)) * Float64(0.005555555555555556 / Float64(180.0 / Float64(angle_m * pi)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = angle_m * (a * pi);
	tmp = 0.0;
	if (angle_m <= 1.5e-77)
		tmp = (b ^ 2.0) + (0.005555555555555556 * (t_0 * (0.005555555555555556 * t_0)));
	else
		tmp = (b ^ 2.0) + (((angle_m * pi) * (a ^ 2.0)) * (0.005555555555555556 / (180.0 / (angle_m * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(a * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle$95$m, 1.5e-77], N[(N[Power[b, 2.0], $MachinePrecision] + N[(0.005555555555555556 * N[(t$95$0 * N[(0.005555555555555556 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.50000000000000008e-77

   1. Initial program 88.2%

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

      1. unpow2 88.2%

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

      2. swap-sqr 88.2%

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

      3. associate-*l* 88.2%

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

      4. associate-*l* 88.2%

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

      5. cancel-sign-sub 88.2%

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

      6. cancel-sign-sub-inv 88.2%

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

      7. associate-*l/ 87.7%

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

      8. associate-/l* 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in angle around 0 88.4%

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

   6. Taylor expanded in angle around 0 83.9%

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

      1. *-commutative 83.9%

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

   8. Simplified 83.9%

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

      1. unpow2 83.9%

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

      2. *-commutative 83.9%

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

      3. associate-*r* 83.9%

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

      4. associate-*l* 84.0%

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

      5. associate-*l* 84.0%

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

   10. Applied egg-rr 84.0%

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

   if 1.50000000000000008e-77 < angle

   1. Initial program 69.8%

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

      1. unpow2 69.8%

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

      2. swap-sqr 69.8%

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

      3. associate-*l* 69.8%

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

      4. associate-*l* 69.8%

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

      5. cancel-sign-sub 69.8%

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

      6. cancel-sign-sub-inv 69.8%

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

      7. associate-*l/ 70.0%

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

      8. associate-/l* 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in angle around 0 69.8%

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

   6. Taylor expanded in angle around 0 62.1%

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

      1. *-commutative 62.1%

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

   8. Simplified 62.1%

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

      1. unpow2 62.1%

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

      2. associate-*r* 62.1%

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

      3. associate-*l* 62.1%

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

      4. *-commutative 62.1%

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

      5. associate-*l* 64.5%

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

      6. *-commutative 64.5%

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

      7. associate-*l* 64.5%

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

      8. associate-*l* 64.5%

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

   10. Applied egg-rr 64.5%

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

       1. *-commutative 64.5%

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

       2. metadata-eval 64.5%

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

       3. associate-/r/ 64.5%

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

       4. un-div-inv 64.6%

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

       5. *-commutative 64.6%

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

       6. associate-*r* 64.6%

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

       7. associate-*r* 64.6%

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

       8. *-commutative 64.6%

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

       9. associate-*l* 64.5%

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

       10. *-commutative 64.5%

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

       11. associate-/r* 64.5%

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

   12. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\frac{\left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right) \cdot 0.005555555555555556}{\frac{\frac{180}{\pi}}{angle}}} + {\left(b \cdot 1\right)}^{2} \]
   13. Step-by-step derivation

       1. associate-/l* 64.5%

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

       2. *-commutative 64.5%

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

       3. associate-*r* 64.6%

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

       4. *-commutative 64.6%

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

       5. associate-*l* 65.0%

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

       6. unpow2 65.0%

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

       7. associate-/l/ 65.0%

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

   14. Simplified 65.0%

       \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right) \cdot \frac{0.005555555555555556}{\frac{180}{angle \cdot \pi}}} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;{b}^{2} + 0.005555555555555556 \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(angle \cdot \pi\right) \cdot {a}^{2}\right) \cdot \frac{0.005555555555555556}{\frac{180}{angle \cdot \pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow b 2.0)
  (*
   (* 0.005555555555555556 (* angle_m PI))
   (* a (* 0.005555555555555556 (* angle_m (* a PI)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(a * N[(0.005555555555555556 * N[(angle$95$m * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. unpow2 82.9%

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

   2. swap-sqr 82.9%

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

   3. associate-*l* 82.9%

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

   4. associate-*l* 82.9%

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

   5. cancel-sign-sub 82.9%

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

   6. cancel-sign-sub-inv 82.9%

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

   7. associate-*l/ 82.6%

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

   8. associate-/l* 83.0%

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

3. Simplified 82.9%

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

5. Taylor expanded in angle around 0 83.0%

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

6. Taylor expanded in angle around 0 77.6%

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

   1. *-commutative 77.6%

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

8. Simplified 77.6%

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

   1. unpow2 77.6%

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

   2. associate-*r* 77.6%

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

   3. associate-*l* 77.6%

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

   4. *-commutative 77.6%

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

   5. associate-*l* 77.5%

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

   6. *-commutative 77.5%

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

   7. associate-*l* 77.5%

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

   8. associate-*l* 77.5%

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

10. Applied egg-rr 77.5%

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

11. Final simplification 77.5%

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

Alternative 6: 74.2% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle_m (* a PI)))))
   (+ (pow b 2.0) (* t_0 t_0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. unpow2 82.9%

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

   2. swap-sqr 82.9%

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

   3. associate-*l* 82.9%

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

   4. associate-*l* 82.9%

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

   5. cancel-sign-sub 82.9%

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

   6. cancel-sign-sub-inv 82.9%

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

   7. associate-*l/ 82.6%

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

   8. associate-/l* 83.0%

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

3. Simplified 82.9%

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

5. Taylor expanded in angle around 0 83.0%

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

6. Taylor expanded in angle around 0 77.6%

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

   1. *-commutative 77.6%

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

8. Simplified 77.6%

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

   1. unpow2 77.6%

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

   2. associate-*l* 77.7%

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

   3. associate-*l* 77.6%

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

10. Applied egg-rr 77.6%

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

11. Final simplification 77.6%

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

Alternative 7: 74.2% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* a PI))))
   (+
    (pow b 2.0)
    (* 0.005555555555555556 (* t_0 (* 0.005555555555555556 t_0))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. unpow2 82.9%

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

   2. swap-sqr 82.9%

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

   3. associate-*l* 82.9%

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

   4. associate-*l* 82.9%

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

   5. cancel-sign-sub 82.9%

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

   6. cancel-sign-sub-inv 82.9%

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

   7. associate-*l/ 82.6%

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

   8. associate-/l* 83.0%

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

3. Simplified 82.9%

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

5. Taylor expanded in angle around 0 83.0%

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

6. Taylor expanded in angle around 0 77.6%

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

   1. *-commutative 77.6%

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

8. Simplified 77.6%

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

   1. unpow2 77.6%

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

   2. *-commutative 77.6%

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

   3. associate-*r* 77.6%

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

   4. associate-*l* 77.7%

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

   5. associate-*l* 77.7%

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

10. Applied egg-rr 77.7%

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

11. Final simplification 77.7%

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

Reproduce

herbie shell --seed 2024046 
(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)))