ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.9%

Time: 43.4s

Alternatives: 10

Speedup: 1.3×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin((sqrt(angle_m) * ((((double) M_PI) * 0.005555555555555556) * sqrt(angle_m))))), 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.sqrt(angle_m) * ((Math.PI * 0.005555555555555556) * Math.sqrt(angle_m))))), 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.sqrt(angle_m) * ((math.pi * 0.005555555555555556) * math.sqrt(angle_m))))), 2.0) + math.pow(b, 2.0)

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin((sqrt(angle_m) * ((pi * 0.005555555555555556) * sqrt(angle_m))))) ^ 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[Sqrt[angle$95$m], $MachinePrecision] * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Sqrt[angle$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. unpow2 76.3%

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

   2. *-commutative 76.3%

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

   3. associate-*r/ 75.9%

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

   4. associate-/l* 76.3%

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

   5. unpow2 76.3%

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

   6. *-commutative 76.3%

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

   7. associate-*r/ 75.9%

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

   8. associate-/l* 76.4%

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

3. Simplified 76.4%

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

   1. associate-/r/ 76.4%

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

   2. add-sqr-sqrt 39.0%

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

   3. associate-*r* 39.0%

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

   4. div-inv 39.0%

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

   5. metadata-eval 39.0%

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

6. Applied egg-rr 39.0%

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

7. Taylor expanded in angle around 0 39.4%

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

8. Final simplification 39.4%

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

Alternative 2: 80.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * log1p(expm1(sin(Float64(Float64(pi * 0.005555555555555556) * angle_m))))) ^ 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[Log[1 + N[(Exp[N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. unpow2 76.3%

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

   2. swap-sqr 76.3%

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

   3. *-commutative 76.3%

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

   4. associate-*r/ 75.9%

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

   5. associate-*l/ 76.4%

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

   6. *-commutative 76.4%

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

   7. swap-sqr 76.4%

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

   8. unpow2 76.4%

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

   9. *-commutative 76.4%

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

   10. associate-*r/ 76.0%

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

   11. associate-*l/ 76.4%

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

   12. *-commutative 76.4%

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

3. Simplified 76.4%

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

5. Taylor expanded in angle around 0 76.5%

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

   1. log1p-expm1-u 76.5%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \frac{\pi}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   2. div-inv 76.5%

      \[\leadsto {\left(a \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \color{blue}{\left(\pi \cdot \frac{1}{180}\right)}\right)\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   3. metadata-eval 76.5%

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

7. Applied egg-rr 76.5%

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

8. Final simplification 76.5%

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

Alternative 3: 80.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin((angle_m * (((double) M_PI) / 180.0)))), 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((angle_m * (Math.PI / 180.0)))), 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((angle_m * (math.pi / 180.0)))), 2.0) + math.pow(b, 2.0)

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin((angle_m * (pi / 180.0)))) ^ 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[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. unpow2 76.3%

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

   2. swap-sqr 76.3%

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

   3. *-commutative 76.3%

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

   4. associate-*r/ 75.9%

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

   5. associate-*l/ 76.4%

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

   6. *-commutative 76.4%

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

   7. swap-sqr 76.4%

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

   8. unpow2 76.4%

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

   9. *-commutative 76.4%

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

   10. associate-*r/ 76.0%

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

   11. associate-*l/ 76.4%

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

   12. *-commutative 76.4%

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

3. Simplified 76.4%

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

5. Taylor expanded in angle around 0 69.1%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. associate-*r* 69.1%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \left(1 + \color{blue}{\left(-1.54320987654321 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot {\pi}^{2}}\right)\right)}^{2} \]

7. Simplified 69.1%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\left(1 + \left(-1.54320987654321 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot {\pi}^{2}\right)}\right)}^{2} \]

8. Taylor expanded in angle around 0 76.5%

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

9. Final simplification 76.5%

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

Alternative 4: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{-128}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(angle\_m \cdot \frac{\pi}{180}\right)\right)}^{2} + \left(\pi \cdot 0.005555555555555556\right) \cdot \left(\left(angle\_m \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 4.1e-128)
   (pow b 2.0)
   (+
    (pow (* b (cos (* angle_m (/ PI 180.0)))) 2.0)
    (*
     (* PI 0.005555555555555556)
     (* (* angle_m (* a (* PI 0.005555555555555556))) (* a angle_m))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.1e-128) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow((b * cos((angle_m * (((double) M_PI) / 180.0)))), 2.0) + ((((double) M_PI) * 0.005555555555555556) * ((angle_m * (a * (((double) M_PI) * 0.005555555555555556))) * (a * angle_m)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.1e-128) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow((b * Math.cos((angle_m * (Math.PI / 180.0)))), 2.0) + ((Math.PI * 0.005555555555555556) * ((angle_m * (a * (Math.PI * 0.005555555555555556))) * (a * angle_m)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 4.1e-128:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow((b * math.cos((angle_m * (math.pi / 180.0)))), 2.0) + ((math.pi * 0.005555555555555556) * ((angle_m * (a * (math.pi * 0.005555555555555556))) * (a * angle_m)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 4.1e-128)
		tmp = b ^ 2.0;
	else
		tmp = Float64((Float64(b * cos(Float64(angle_m * Float64(pi / 180.0)))) ^ 2.0) + Float64(Float64(pi * 0.005555555555555556) * Float64(Float64(angle_m * Float64(a * Float64(pi * 0.005555555555555556))) * Float64(a * angle_m))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 4.1e-128)
		tmp = b ^ 2.0;
	else
		tmp = ((b * cos((angle_m * (pi / 180.0)))) ^ 2.0) + ((pi * 0.005555555555555556) * ((angle_m * (a * (pi * 0.005555555555555556))) * (a * angle_m)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 4.1e-128], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[N[(b * N[Cos[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(N[(angle$95$m * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.1e-128

   1. Initial program 76.1%

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

      1. unpow2 76.1%

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

      2. swap-sqr 76.1%

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

      3. *-commutative 76.1%

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

      4. associate-*r/ 75.6%

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

      5. associate-*l/ 76.2%

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

      6. *-commutative 76.2%

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

      7. swap-sqr 76.1%

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

      8. unpow2 76.1%

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

      9. *-commutative 76.1%

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

      10. associate-*r/ 75.6%

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

      11. associate-*l/ 76.3%

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

      12. *-commutative 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in angle around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*r* 68.3%

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

      3. associate-*l* 68.3%

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

      4. *-commutative 68.3%

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

      5. *-commutative 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in angle around 0 67.9%

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

   9. Taylor expanded in angle around 0 58.3%

      \[\leadsto \color{blue}{{b}^{2}} \]

   if 4.1e-128 < a

   1. Initial program 76.6%

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

      1. unpow2 76.6%

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

      2. swap-sqr 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*r/ 76.6%

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

      5. associate-*l/ 76.8%

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

      6. *-commutative 76.8%

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

      7. swap-sqr 76.8%

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

      8. unpow2 76.8%

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

      9. *-commutative 76.8%

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

      10. associate-*r/ 76.8%

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

      11. associate-*l/ 76.8%

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

      12. *-commutative 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in angle around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-*r* 71.4%

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

      3. associate-*l* 71.4%

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

      4. *-commutative 71.4%

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

      5. *-commutative 71.4%

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

   7. Simplified 71.4%

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

      1. unpow2 71.4%

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

      2. *-commutative 71.4%

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

      3. metadata-eval 71.4%

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

      4. div-inv 71.4%

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

      5. associate-*r* 71.5%

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

      6. associate-*l* 71.4%

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

      7. *-commutative 71.4%

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

      8. div-inv 71.4%

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

      9. metadata-eval 71.4%

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

   9. Applied egg-rr 71.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{-128}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + \left(\pi \cdot 0.005555555555555556\right) \cdot \left(\left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(angle\_m \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot angle\_m\right) + {\left(b \cdot \cos \left(angle\_m \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.1e-128)
   (pow b 2.0)
   (+
    (*
     (*
      (* PI 0.005555555555555556)
      (* angle_m (* a (* PI 0.005555555555555556))))
     (* a angle_m))
    (pow (* b (cos (* angle_m (/ PI 180.0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.1e-128) {
		tmp = pow(b, 2.0);
	} else {
		tmp = (((((double) M_PI) * 0.005555555555555556) * (angle_m * (a * (((double) M_PI) * 0.005555555555555556)))) * (a * angle_m)) + pow((b * cos((angle_m * (((double) M_PI) / 180.0)))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.1e-128) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = (((Math.PI * 0.005555555555555556) * (angle_m * (a * (Math.PI * 0.005555555555555556)))) * (a * angle_m)) + Math.pow((b * Math.cos((angle_m * (Math.PI / 180.0)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.1e-128:
		tmp = math.pow(b, 2.0)
	else:
		tmp = (((math.pi * 0.005555555555555556) * (angle_m * (a * (math.pi * 0.005555555555555556)))) * (a * angle_m)) + math.pow((b * math.cos((angle_m * (math.pi / 180.0)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.1e-128)
		tmp = b ^ 2.0;
	else
		tmp = Float64(Float64(Float64(Float64(pi * 0.005555555555555556) * Float64(angle_m * Float64(a * Float64(pi * 0.005555555555555556)))) * Float64(a * angle_m)) + (Float64(b * cos(Float64(angle_m * Float64(pi / 180.0)))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.1e-128)
		tmp = b ^ 2.0;
	else
		tmp = (((pi * 0.005555555555555556) * (angle_m * (a * (pi * 0.005555555555555556)))) * (a * angle_m)) + ((b * cos((angle_m * (pi / 180.0)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.1e-128], N[Power[b, 2.0], $MachinePrecision], N[(N[(N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(angle$95$m * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Cos[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.10000000000000005e-128

   1. Initial program 76.1%

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

      1. unpow2 76.1%

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

      2. swap-sqr 76.1%

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

      3. *-commutative 76.1%

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

      4. associate-*r/ 75.6%

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

      5. associate-*l/ 76.2%

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

      6. *-commutative 76.2%

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

      7. swap-sqr 76.1%

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

      8. unpow2 76.1%

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

      9. *-commutative 76.1%

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

      10. associate-*r/ 75.6%

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

      11. associate-*l/ 76.3%

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

      12. *-commutative 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in angle around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*r* 68.3%

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

      3. associate-*l* 68.3%

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

      4. *-commutative 68.3%

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

      5. *-commutative 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in angle around 0 67.9%

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

   9. Taylor expanded in angle around 0 58.3%

      \[\leadsto \color{blue}{{b}^{2}} \]

   if 1.10000000000000005e-128 < a

   1. Initial program 76.6%

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

      1. unpow2 76.6%

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

      2. swap-sqr 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*r/ 76.6%

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

      5. associate-*l/ 76.8%

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

      6. *-commutative 76.8%

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

      7. swap-sqr 76.8%

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

      8. unpow2 76.8%

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

      9. *-commutative 76.8%

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

      10. associate-*r/ 76.8%

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

      11. associate-*l/ 76.8%

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

      12. *-commutative 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in angle around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-*r* 71.4%

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

      3. associate-*l* 71.4%

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

      4. *-commutative 71.4%

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

      5. *-commutative 71.4%

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

   7. Simplified 71.4%

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

      1. unpow2 71.5%

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

      2. *-commutative 71.5%

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

      3. metadata-eval 71.5%

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

      4. div-inv 71.5%

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

      5. *-commutative 71.5%

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

      6. associate-*r* 71.5%

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

      7. associate-*l* 71.5%

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

      8. *-commutative 71.5%

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

      9. div-inv 71.5%

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

      10. metadata-eval 71.5%

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

   9. Applied egg-rr 71.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot angle\right) + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + angle\_m \cdot \left(\left(a \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(a \cdot angle\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 2e-64)
   (pow b 2.0)
   (+
    (pow b 2.0)
    (*
     angle_m
     (*
      (* a 0.005555555555555556)
      (* PI (* (* PI 0.005555555555555556) (* a angle_m))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2e-64) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + (angle_m * ((a * 0.005555555555555556) * (((double) M_PI) * ((((double) M_PI) * 0.005555555555555556) * (a * angle_m)))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2e-64) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (angle_m * ((a * 0.005555555555555556) * (Math.PI * ((Math.PI * 0.005555555555555556) * (a * angle_m)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 2e-64:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + (angle_m * ((a * 0.005555555555555556) * (math.pi * ((math.pi * 0.005555555555555556) * (a * angle_m)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 2e-64)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(angle_m * Float64(Float64(a * 0.005555555555555556) * Float64(pi * Float64(Float64(pi * 0.005555555555555556) * Float64(a * angle_m))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 2e-64)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (angle_m * ((a * 0.005555555555555556) * (pi * ((pi * 0.005555555555555556) * (a * angle_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 2e-64], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(a * 0.005555555555555556), $MachinePrecision] * N[(Pi * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.99999999999999993e-64

   1. Initial program 75.6%

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

      1. unpow2 75.6%

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

      2. swap-sqr 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-*r/ 75.2%

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

      5. associate-*l/ 75.7%

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

      6. *-commutative 75.7%

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

      7. swap-sqr 75.7%

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

      8. unpow2 75.7%

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

      9. *-commutative 75.7%

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

      10. associate-*r/ 75.2%

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

      11. associate-*l/ 75.8%

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

      12. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*r* 67.6%

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

      3. associate-*l* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in angle around 0 67.2%

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

   9. Taylor expanded in angle around 0 58.3%

      \[\leadsto \color{blue}{{b}^{2}} \]

   if 1.99999999999999993e-64 < a

   1. Initial program 77.9%

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

      1. unpow2 77.9%

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

      2. swap-sqr 77.9%

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

      3. *-commutative 77.9%

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

      4. associate-*r/ 77.9%

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

      5. associate-*l/ 78.1%

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

      6. *-commutative 78.1%

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

      7. swap-sqr 78.1%

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

      8. unpow2 78.1%

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

      9. *-commutative 78.1%

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

      10. associate-*r/ 78.1%

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

      11. associate-*l/ 78.1%

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

      12. *-commutative 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in angle around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*r* 73.6%

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

      3. associate-*l* 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in angle around 0 73.8%

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

      1. unpow2 73.8%

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

      2. associate-*r* 73.7%

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

      3. associate-*l* 73.8%

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

      4. associate-*l* 73.7%

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

      5. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

   10. Applied egg-rr 73.7%

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

       1. associate-*l* 69.9%

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

       2. *-commutative 69.9%

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

       3. associate-*r* 69.9%

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

       4. *-commutative 69.9%

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

       5. *-commutative 69.9%

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

   12. Simplified 69.9%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + angle \cdot \left(\left(a \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(a \cdot angle\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 5.3 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + t\_0 \cdot \left(angle\_m \cdot \left(angle\_m \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* a (* PI 0.005555555555555556))))
   (if (<= a 5.3e-64)
     (pow b 2.0)
     (+ (pow b 2.0) (* t_0 (* angle_m (* angle_m t_0)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = a * (((double) M_PI) * 0.005555555555555556);
	double tmp;
	if (a <= 5.3e-64) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + (t_0 * (angle_m * (angle_m * t_0)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = a * (Math.PI * 0.005555555555555556);
	double tmp;
	if (a <= 5.3e-64) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (t_0 * (angle_m * (angle_m * t_0)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = a * (math.pi * 0.005555555555555556)
	tmp = 0
	if a <= 5.3e-64:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + (t_0 * (angle_m * (angle_m * t_0)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(a * Float64(pi * 0.005555555555555556))
	tmp = 0.0
	if (a <= 5.3e-64)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(t_0 * Float64(angle_m * Float64(angle_m * t_0))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = a * (pi * 0.005555555555555556);
	tmp = 0.0;
	if (a <= 5.3e-64)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (t_0 * (angle_m * (angle_m * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.3e-64], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$0 * N[(angle$95$m * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.3000000000000002e-64

   1. Initial program 75.6%

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

      1. unpow2 75.6%

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

      2. swap-sqr 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-*r/ 75.2%

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

      5. associate-*l/ 75.7%

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

      6. *-commutative 75.7%

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

      7. swap-sqr 75.7%

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

      8. unpow2 75.7%

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

      9. *-commutative 75.7%

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

      10. associate-*r/ 75.2%

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

      11. associate-*l/ 75.8%

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

      12. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*r* 67.6%

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

      3. associate-*l* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in angle around 0 67.2%

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

   9. Taylor expanded in angle around 0 58.3%

      \[\leadsto \color{blue}{{b}^{2}} \]

   if 5.3000000000000002e-64 < a

   1. Initial program 77.9%

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

      1. unpow2 77.9%

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

      2. swap-sqr 77.9%

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

      3. *-commutative 77.9%

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

      4. associate-*r/ 77.9%

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

      5. associate-*l/ 78.1%

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

      6. *-commutative 78.1%

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

      7. swap-sqr 78.1%

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

      8. unpow2 78.1%

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

      9. *-commutative 78.1%

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

      10. associate-*r/ 78.1%

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

      11. associate-*l/ 78.1%

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

      12. *-commutative 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in angle around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*r* 73.6%

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

      3. associate-*l* 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in angle around 0 73.8%

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

      1. unpow2 73.8%

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

      2. associate-*l* 73.7%

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

      3. associate-*r* 72.4%

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

      4. associate-*l* 72.4%

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

      5. *-commutative 72.4%

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

      6. *-commutative 72.4%

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

   10. Applied egg-rr 72.4%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.3 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \pi \cdot \left(\left(angle\_m \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(angle\_m \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.8e-64)
   (pow b 2.0)
   (+
    (pow b 2.0)
    (*
     PI
     (*
      (* angle_m (* a (* PI 0.005555555555555556)))
      (* angle_m (* a 0.005555555555555556)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.8e-64) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + (((double) M_PI) * ((angle_m * (a * (((double) M_PI) * 0.005555555555555556))) * (angle_m * (a * 0.005555555555555556))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.8e-64) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (Math.PI * ((angle_m * (a * (Math.PI * 0.005555555555555556))) * (angle_m * (a * 0.005555555555555556))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3.8e-64:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + (math.pi * ((angle_m * (a * (math.pi * 0.005555555555555556))) * (angle_m * (a * 0.005555555555555556))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.8e-64)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(pi * Float64(Float64(angle_m * Float64(a * Float64(pi * 0.005555555555555556))) * Float64(angle_m * Float64(a * 0.005555555555555556)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 3.8e-64)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (pi * ((angle_m * (a * (pi * 0.005555555555555556))) * (angle_m * (a * 0.005555555555555556))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.8e-64], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(Pi * N[(N[(angle$95$m * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.8000000000000002e-64

   1. Initial program 75.6%

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

      1. unpow2 75.6%

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

      2. swap-sqr 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-*r/ 75.2%

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

      5. associate-*l/ 75.7%

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

      6. *-commutative 75.7%

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

      7. swap-sqr 75.7%

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

      8. unpow2 75.7%

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

      9. *-commutative 75.7%

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

      10. associate-*r/ 75.2%

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

      11. associate-*l/ 75.8%

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

      12. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*r* 67.6%

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

      3. associate-*l* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in angle around 0 67.2%

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

   9. Taylor expanded in angle around 0 58.3%

      \[\leadsto \color{blue}{{b}^{2}} \]

   if 3.8000000000000002e-64 < a

   1. Initial program 77.9%

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

      1. unpow2 77.9%

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

      2. swap-sqr 77.9%

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

      3. *-commutative 77.9%

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

      4. associate-*r/ 77.9%

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

      5. associate-*l/ 78.1%

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

      6. *-commutative 78.1%

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

      7. swap-sqr 78.1%

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

      8. unpow2 78.1%

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

      9. *-commutative 78.1%

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

      10. associate-*r/ 78.1%

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

      11. associate-*l/ 78.1%

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

      12. *-commutative 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in angle around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*r* 73.6%

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

      3. associate-*l* 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in angle around 0 73.8%

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

      1. unpow2 73.8%

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

      2. associate-*r* 73.7%

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

      3. associate-*r* 73.7%

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

      4. associate-*l* 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*l* 73.7%

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

   10. Applied egg-rr 73.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \pi \cdot \left(\left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.18 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(angle\_m \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.18e-64)
   (pow b 2.0)
   (+
    (pow b 2.0)
    (*
     (*
      (* PI 0.005555555555555556)
      (* angle_m (* a (* PI 0.005555555555555556))))
     (* a angle_m)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.18e-64) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + (((((double) M_PI) * 0.005555555555555556) * (angle_m * (a * (((double) M_PI) * 0.005555555555555556)))) * (a * angle_m));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.18e-64) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (((Math.PI * 0.005555555555555556) * (angle_m * (a * (Math.PI * 0.005555555555555556)))) * (a * angle_m));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.18e-64:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + (((math.pi * 0.005555555555555556) * (angle_m * (a * (math.pi * 0.005555555555555556)))) * (a * angle_m))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.18e-64)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(Float64(pi * 0.005555555555555556) * Float64(angle_m * Float64(a * Float64(pi * 0.005555555555555556)))) * Float64(a * angle_m)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.18e-64)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (((pi * 0.005555555555555556) * (angle_m * (a * (pi * 0.005555555555555556)))) * (a * angle_m));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.18e-64], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(angle$95$m * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.17999999999999996e-64

   1. Initial program 75.6%

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

      1. unpow2 75.6%

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

      2. swap-sqr 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-*r/ 75.2%

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

      5. associate-*l/ 75.7%

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

      6. *-commutative 75.7%

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

      7. swap-sqr 75.7%

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

      8. unpow2 75.7%

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

      9. *-commutative 75.7%

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

      10. associate-*r/ 75.2%

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

      11. associate-*l/ 75.8%

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

      12. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in angle around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*r* 67.6%

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

      3. associate-*l* 67.6%

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

      4. *-commutative 67.6%

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

      5. *-commutative 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in angle around 0 67.2%

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

   9. Taylor expanded in angle around 0 58.3%

      \[\leadsto \color{blue}{{b}^{2}} \]

   if 1.17999999999999996e-64 < a

   1. Initial program 77.9%

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

      1. unpow2 77.9%

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

      2. swap-sqr 77.9%

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

      3. *-commutative 77.9%

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

      4. associate-*r/ 77.9%

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

      5. associate-*l/ 78.1%

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

      6. *-commutative 78.1%

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

      7. swap-sqr 78.1%

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

      8. unpow2 78.1%

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

      9. *-commutative 78.1%

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

      10. associate-*r/ 78.1%

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

      11. associate-*l/ 78.1%

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

      12. *-commutative 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in angle around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*r* 73.6%

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

      3. associate-*l* 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in angle around 0 73.8%

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

      1. unpow2 73.8%

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

      2. *-commutative 73.8%

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

      3. metadata-eval 73.8%

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

      4. div-inv 73.8%

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

      5. *-commutative 73.8%

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

      6. associate-*r* 73.8%

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

      7. associate-*l* 73.7%

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

      8. *-commutative 73.7%

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

      9. div-inv 73.7%

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

      10. metadata-eval 73.7%

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

   10. Applied egg-rr 73.7%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.18 \cdot 10^{-64}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot angle\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {b}^{2} \end{array} \]

FPCore

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

C

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

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = b ** 2.0d0
end function

Java

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return b ^ 2.0
end

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[Power[b, 2.0], $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. unpow2 76.3%

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

   2. swap-sqr 76.3%

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

   3. *-commutative 76.3%

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

   4. associate-*r/ 75.9%

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

   5. associate-*l/ 76.4%

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

   6. *-commutative 76.4%

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

   7. swap-sqr 76.4%

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

   8. unpow2 76.4%

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

   9. *-commutative 76.4%

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

   10. associate-*r/ 76.0%

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

   11. associate-*l/ 76.4%

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

   12. *-commutative 76.4%

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

3. Simplified 76.4%

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

5. Taylor expanded in angle around 0 69.3%

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

   1. *-commutative 69.3%

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

   2. associate-*r* 69.3%

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

   3. associate-*l* 69.3%

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

   4. *-commutative 69.3%

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

   5. *-commutative 69.3%

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

7. Simplified 69.3%

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

8. Taylor expanded in angle around 0 69.0%

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

9. Taylor expanded in angle around 0 51.3%

   \[\leadsto \color{blue}{{b}^{2}} \]

10. Final simplification 51.3%

    \[\leadsto {b}^{2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))