ab-angle->ABCF C

Percentage Accurate: 80.0% → 80.0%

Time: 14.7s

Alternatives: 10

Speedup: 1.0×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. clear-num 81.1%

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

   4. un-div-inv 81.2%

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

6. Applied egg-rr 81.2%

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

   1. div-inv 81.1%

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

   2. clear-num 81.2%

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

   3. div-inv 81.2%

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

   4. metadata-eval 81.2%

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

   5. add-sqr-sqrt 37.8%

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

   6. associate-*r* 37.9%

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

8. Applied egg-rr 37.9%

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

9. Final simplification 37.9%

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

Alternative 2: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin (expm1 (log1p t_0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	return pow((a * cos(t_0)), 2.0) + pow((b * sin(expm1(log1p(t_0)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	return Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(Math.expm1(Math.log1p(t_0)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	return math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(math.expm1(math.log1p(t_0)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	return Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(expm1(log1p(t_0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. clear-num 81.1%

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

   4. un-div-inv 81.2%

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

6. Applied egg-rr 81.2%

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

   1. expm1-log1p-u 64.9%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)}\right)}^{2} \]

   2. expm1-undefine 55.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{\frac{180}{angle}}\right)} - 1\right)}\right)}^{2} \]

   3. associate-/r/ 55.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\pi}{180} \cdot angle}\right)} - 1\right)\right)}^{2} \]

   4. div-inv 55.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot \frac{1}{180}\right)} \cdot angle\right)} - 1\right)\right)}^{2} \]

   5. metadata-eval 55.5%

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

   6. *-commutative 55.5%

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

   7. *-commutative 55.5%

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

   8. *-commutative 55.5%

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

8. Applied egg-rr 55.5%

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

   1. expm1-define 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 64.9%

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

10. Simplified 64.9%

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

11. Final simplification 64.9%

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

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(\mathsf{hypot}\left(a \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(\frac{\pi}{180 \cdot \frac{1}{angle\_m}}\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (pow
  (hypot
   (* a (cos (* PI (* angle_m 0.005555555555555556))))
   (* b (sin (/ PI (* 180.0 (/ 1.0 angle_m))))))
  2.0))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(hypot((a * cos((((double) M_PI) * (angle_m * 0.005555555555555556)))), (b * sin((((double) M_PI) / (180.0 * (1.0 / angle_m)))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow(Math.hypot((a * Math.cos((Math.PI * (angle_m * 0.005555555555555556)))), (b * Math.sin((Math.PI / (180.0 * (1.0 / angle_m)))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow(math.hypot((a * math.cos((math.pi * (angle_m * 0.005555555555555556)))), (b * math.sin((math.pi / (180.0 * (1.0 / angle_m)))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return hypot(Float64(a * cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))), Float64(b * sin(Float64(pi / Float64(180.0 * Float64(1.0 / angle_m)))))) ^ 2.0
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = hypot((a * cos((pi * (angle_m * 0.005555555555555556)))), (b * sin((pi / (180.0 * (1.0 / angle_m)))))) ^ 2.0;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[Power[N[Sqrt[N[(a * N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[N[(Pi / N[(180.0 * N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. clear-num 81.1%

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

   4. un-div-inv 81.2%

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

6. Applied egg-rr 81.2%

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

   1. add-sqr-sqrt 81.1%

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

   2. div-inv 81.1%

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

   3. times-frac 81.1%

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

8. Applied egg-rr 81.1%

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

10. Applied egg-rr 81.2%

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(\frac{\pi}{180 \cdot \frac{1}{angle}}\right)\right)\right)}^{2}} \]
11. Add Preprocessing

Alternative 4: 80.0% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[Power[N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. clear-num 81.1%

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

   4. un-div-inv 81.2%

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

6. Applied egg-rr 81.2%

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

   1. div-inv 81.1%

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

   2. clear-num 81.2%

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

   3. div-inv 81.2%

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

   4. metadata-eval 81.2%

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

   5. add-sqr-sqrt 37.8%

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

   6. associate-*r* 37.9%

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

8. Applied egg-rr 37.9%

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

10. Applied egg-rr 81.2%

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2}} \]
11. Add Preprocessing

Alternative 6: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot \left(a \cdot \frac{1 + \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)}{2}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (* a (* a (/ (+ 1.0 (cos (* PI (* angle_m 0.011111111111111112)))) 2.0)))
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return (a * (a * ((1.0 + cos((((double) M_PI) * (angle_m * 0.011111111111111112)))) / 2.0))) + pow((b * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return (a * (a * ((1.0 + Math.cos((Math.PI * (angle_m * 0.011111111111111112)))) / 2.0))) + Math.pow((b * Math.sin((Math.PI * (angle_m * 0.005555555555555556)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return (a * (a * ((1.0 + math.cos((math.pi * (angle_m * 0.011111111111111112)))) / 2.0))) + math.pow((b * math.sin((math.pi * (angle_m * 0.005555555555555556)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(a * Float64(a * Float64(Float64(1.0 + cos(Float64(pi * Float64(angle_m * 0.011111111111111112)))) / 2.0))) + (Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a * (a * ((1.0 + cos((pi * (angle_m * 0.011111111111111112)))) / 2.0))) + ((b * sin((pi * (angle_m * 0.005555555555555556)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(a * N[(a * N[(N[(1.0 + N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. unpow2 81.2%

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

   4. *-commutative 81.2%

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

   5. associate-*r* 81.2%

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

6. Applied egg-rr 81.1%

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

7. Taylor expanded in a around 0 81.0%

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

   1. unpow2 81.0%

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

   2. associate-*r* 80.0%

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

   3. *-commutative 80.0%

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

   4. *-commutative 80.0%

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

   5. associate-*r* 81.2%

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

   6. *-commutative 81.2%

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

   7. *-commutative 81.2%

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

   8. cos-mult 81.2%

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

9. Applied egg-rr 81.2%

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

    1. +-commutative 81.2%

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

    2. +-inverses 81.2%

       \[\leadsto \left(a \cdot \frac{\cos \color{blue}{0} + \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{2}\right) \cdot a + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    3. cos-0 81.2%

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

    4. distribute-lft-out 81.2%

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

    5. distribute-lft-out 81.2%

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

    6. metadata-eval 81.2%

       \[\leadsto \left(a \cdot \frac{1 + \cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)}{2}\right) \cdot a + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

11. Simplified 81.2%

    \[\leadsto \left(a \cdot \color{blue}{\frac{1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{2}}\right) \cdot a + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

12. Final simplification 81.2%

    \[\leadsto a \cdot \left(a \cdot \frac{1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{2}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
13. Add Preprocessing

Alternative 7: 53.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{-34}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 7.8e-34)
   (pow (* b (sin (* angle_m (* PI 0.005555555555555556)))) 2.0)
   (pow (* a (cos (* PI (* angle_m 0.005555555555555556)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 7.8e-34) {
		tmp = pow((b * sin((angle_m * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	} else {
		tmp = pow((a * cos((((double) M_PI) * (angle_m * 0.005555555555555556)))), 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 <= 7.8e-34) {
		tmp = Math.pow((b * Math.sin((angle_m * (Math.PI * 0.005555555555555556)))), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((Math.PI * (angle_m * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 7.8e-34:
		tmp = math.pow((b * math.sin((angle_m * (math.pi * 0.005555555555555556)))), 2.0)
	else:
		tmp = math.pow((a * math.cos((math.pi * (angle_m * 0.005555555555555556)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 7.8e-34)
		tmp = Float64(b * sin(Float64(angle_m * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = Float64(a * cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 7.8e-34)
		tmp = (b * sin((angle_m * (pi * 0.005555555555555556)))) ^ 2.0;
	else
		tmp = (a * cos((pi * (angle_m * 0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 7.8e-34], N[Power[N[(b * N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(a * N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.79999999999999982e-34

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

      1. metadata-eval 79.5%

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

      2. div-inv 79.6%

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

      3. clear-num 79.4%

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

      4. un-div-inv 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. div-inv 79.4%

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

      2. clear-num 79.6%

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

      3. div-inv 79.5%

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

      4. metadata-eval 79.5%

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

      5. add-sqr-sqrt 37.5%

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

      6. associate-*r* 37.7%

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

   8. Applied egg-rr 37.7%

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

   9. Taylor expanded in a around 0 45.3%

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

       1. unpow2 45.3%

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

       2. associate-*r* 45.2%

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

       3. unpow2 45.2%

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

       4. rem-square-sqrt 45.2%

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

       5. *-commutative 45.2%

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

       6. *-commutative 45.2%

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

       7. *-commutative 45.2%

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

       8. associate-*r* 45.2%

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

       9. unpow2 45.2%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

       10. swap-sqr 49.2%

           \[\leadsto \color{blue}{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \]

       11. unpow2 49.2%

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

   11. Simplified 49.2%

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

   if 7.79999999999999982e-34 < a

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. metadata-eval 87.1%

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

      2. div-inv 87.1%

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

      3. clear-num 87.1%

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

      4. un-div-inv 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in a around inf 78.2%

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

      1. unpow2 78.2%

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

      2. *-commutative 78.2%

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

      3. associate-*r* 78.2%

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

      4. *-commutative 78.2%

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

      5. *-commutative 78.2%

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

      6. unpow2 78.2%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \]

      7. swap-sqr 76.8%

         \[\leadsto \color{blue}{\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \]

      8. unpow2 76.8%

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

   9. Simplified 78.2%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.8 \cdot 10^{-34}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-33}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 7.2e-33)
   (pow (* b (sin (* 0.005555555555555556 (* PI angle_m)))) 2.0)
   (pow (* a (cos (* PI (* angle_m 0.005555555555555556)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 7.2e-33) {
		tmp = pow((b * sin((0.005555555555555556 * (((double) M_PI) * angle_m)))), 2.0);
	} else {
		tmp = pow((a * cos((((double) M_PI) * (angle_m * 0.005555555555555556)))), 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 <= 7.2e-33) {
		tmp = Math.pow((b * Math.sin((0.005555555555555556 * (Math.PI * angle_m)))), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((Math.PI * (angle_m * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 7.2e-33:
		tmp = math.pow((b * math.sin((0.005555555555555556 * (math.pi * angle_m)))), 2.0)
	else:
		tmp = math.pow((a * math.cos((math.pi * (angle_m * 0.005555555555555556)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 7.2e-33)
		tmp = Float64(b * sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))) ^ 2.0;
	else
		tmp = Float64(a * cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 7.2e-33)
		tmp = (b * sin((0.005555555555555556 * (pi * angle_m)))) ^ 2.0;
	else
		tmp = (a * cos((pi * (angle_m * 0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 7.2e-33], N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(a * N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.20000000000000068e-33

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in a around 0 45.2%

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

      1. unpow2 45.2%

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

      2. *-commutative 45.2%

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

      3. unpow2 45.2%

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      4. swap-sqr 49.1%

         \[\leadsto \color{blue}{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      5. unpow2 49.1%

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

      6. *-commutative 49.1%

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

   7. Simplified 49.1%

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

   if 7.20000000000000068e-33 < a

   1. Initial program 87.1%

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

   3. Simplified 87.1%

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

      1. metadata-eval 87.1%

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

      2. div-inv 87.1%

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

      3. clear-num 87.1%

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

      4. un-div-inv 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in a around inf 78.2%

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

      1. unpow2 78.2%

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

      2. *-commutative 78.2%

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

      3. associate-*r* 78.2%

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

      4. *-commutative 78.2%

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

      5. *-commutative 78.2%

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

      6. unpow2 78.2%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \]

      7. swap-sqr 76.8%

         \[\leadsto \color{blue}{\left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \]

      8. unpow2 76.8%

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

   9. Simplified 78.2%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-33}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. unpow2 81.2%

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

   4. *-commutative 81.2%

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

   5. associate-*r* 81.2%

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

6. Applied egg-rr 81.1%

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

7. Taylor expanded in angle around 0 81.0%

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

8. Final simplification 81.0%

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

Alternative 10: 56.9% accurate, 139.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

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

C

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

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 = a * a
end function

Java

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(a * a)
end

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 60.5%

   \[\leadsto \color{blue}{{a}^{2}} \]
6. Step-by-step derivation

   1. unpow2 60.5%

      \[\leadsto \color{blue}{a \cdot a} \]

7. Applied egg-rr 60.5%

   \[\leadsto \color{blue}{a \cdot a} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))