ab-angle->ABCF A

Percentage Accurate: 80.4% → 80.4%

Time: 18.2s

Alternatives: 8

Speedup: 1.0×

• 36.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (expm1 (log1p (* angle_m (* PI 0.005555555555555556)))))) 2.0)
  (pow (* b (cos (* PI (/ angle_m 180.0)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(expm1(log1p((angle_m * (((double) M_PI) * 0.005555555555555556)))))), 2.0) + pow((b * cos((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin(Math.expm1(Math.log1p((angle_m * (Math.PI * 0.005555555555555556)))))), 2.0) + Math.pow((b * Math.cos((Math.PI * (angle_m / 180.0)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin(math.expm1(math.log1p((angle_m * (math.pi * 0.005555555555555556)))))), 2.0) + math.pow((b * math.cos((math.pi * (angle_m / 180.0)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(expm1(log1p(Float64(angle_m * Float64(pi * 0.005555555555555556)))))) ^ 2.0) + (Float64(b * cos(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(Exp[N[Log[1 + N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. expm1-log1p-u 63.2%

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

   2. expm1-undefine 52.0%

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

   3. associate-*l/ 52.0%

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

   4. associate-*r/ 52.0%

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

   5. div-inv 52.0%

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

   6. metadata-eval 52.0%

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

4. Applied egg-rr 52.0%

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

   1. expm1-define 63.2%

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

6. Simplified 63.2%

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

7. Final simplification 63.2%

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

Alternative 2: 80.4% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (+ (pow (* b (cos t_0)) 2.0) (pow (* a (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((b * cos(t_0)), 2.0) + pow((a * 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((b * Math.cos(t_0)), 2.0) + Math.pow((a * 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((b * math.cos(t_0)), 2.0) + math.pow((a * 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(b * cos(t_0)) ^ 2.0) + (Float64(a * 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 = ((b * cos(t_0)) ^ 2.0) + ((a * 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[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.0%

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

3. Final simplification 81.0%

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

Alternative 3: 80.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $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]]

Derivation

1. Initial program 81.0%

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

   1. associate-*l/ 81.0%

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

   2. associate-/l* 81.0%

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

   3. cos-neg 81.0%

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

   4. distribute-lft-neg-out 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-frac-neg 81.0%

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

   7. distribute-lft-neg-out 81.0%

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

   8. cos-neg 81.0%

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

   9. associate-*l/ 80.9%

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

   10. associate-/l* 80.9%

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

3. Simplified 80.9%

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

Alternative 4: 80.4% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (pow (hypot (* a (sin t_0)) (* b (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((a * sin(t_0)), (b * 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((a * Math.sin(t_0)), (b * 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((a * math.sin(t_0)), (b * 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(a * sin(t_0)), Float64(b * 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((a * sin(t_0)), (b * 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[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Initial program 81.0%

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

   1. associate-*l/ 81.0%

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

   2. associate-/l* 81.0%

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

   3. cos-neg 81.0%

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

   4. distribute-lft-neg-out 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-frac-neg 81.0%

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

   7. distribute-lft-neg-out 81.0%

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

   8. cos-neg 81.0%

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

   9. associate-*l/ 80.9%

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

   10. associate-/l* 80.9%

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

3. Simplified 80.9%

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

6. Applied egg-rr 51.5%

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

8. Applied egg-rr 80.9%

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

9. Final simplification 80.9%

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

Alternative 5: 80.2% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (* PI (/ angle_m 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((((double) M_PI) * (angle_m / 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((Math.PI * (angle_m / 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((math.pi * (angle_m / 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(pi * Float64(angle_m / 180.0)))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

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

Derivation

1. Initial program 81.0%

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

   1. associate-*l/ 80.9%

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

   2. clear-num 80.9%

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

4. Applied egg-rr 80.9%

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

5. Taylor expanded in angle around 0 80.9%

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

6. Final simplification 80.9%

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

Alternative 6: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.2e+148)
   (* b b)
   (pow (* a (sin (* angle_m (* PI 0.005555555555555556)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.2e+148) {
		tmp = b * b;
	} else {
		tmp = pow((a * sin((angle_m * (((double) M_PI) * 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 <= 3.2e+148) {
		tmp = b * b;
	} else {
		tmp = Math.pow((a * Math.sin((angle_m * (Math.PI * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3.2e+148:
		tmp = b * b
	else:
		tmp = math.pow((a * math.sin((angle_m * (math.pi * 0.005555555555555556)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.2e+148)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * sin(Float64(angle_m * Float64(pi * 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 <= 3.2e+148)
		tmp = b * b;
	else
		tmp = (a * sin((angle_m * (pi * 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, 3.2e+148], N[(b * b), $MachinePrecision], N[Power[N[(a * N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.1999999999999999e148

   1. Initial program 78.5%

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

      1. associate-*l/ 78.5%

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

      2. associate-/l* 78.4%

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

      3. cos-neg 78.4%

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

      4. distribute-lft-neg-out 78.4%

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

      5. distribute-frac-neg 78.4%

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

      6. distribute-frac-neg 78.4%

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

      7. distribute-lft-neg-out 78.4%

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

      8. cos-neg 78.4%

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

      9. associate-*l/ 78.4%

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

      10. associate-/l* 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in angle around 0 63.1%

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

      1. unpow2 63.1%

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

   7. Applied egg-rr 63.1%

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

   if 3.1999999999999999e148 < a

   1. Initial program 97.4%

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

      1. associate-*l/ 97.3%

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

      2. associate-/l* 97.5%

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

      3. cos-neg 97.5%

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

      4. distribute-lft-neg-out 97.5%

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

      5. distribute-frac-neg 97.5%

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

      6. distribute-frac-neg 97.5%

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

      7. distribute-lft-neg-out 97.5%

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

      8. cos-neg 97.5%

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

      9. associate-*l/ 97.5%

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

      10. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

   6. Applied egg-rr 66.8%

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

   8. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{{\left({\left({\left(\mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot b\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. Taylor expanded in a around -inf 71.8%

      \[\leadsto {\left({\left({\color{blue}{\left(-1 \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]
   10. Step-by-step derivation

       1. mul-1-neg 71.8%

          \[\leadsto {\left({\left({\color{blue}{\left(-a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       2. associate-*r* 71.8%

          \[\leadsto {\left({\left({\left(-a \cdot \sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       3. *-commutative 71.8%

          \[\leadsto {\left({\left({\left(-a \cdot \sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       4. associate-*l* 71.8%

          \[\leadsto {\left({\left({\left(-a \cdot \sin \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       5. *-commutative 71.8%

          \[\leadsto {\left({\left({\left(-a \cdot \sin \left(angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)}\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       6. distribute-rgt-neg-in 71.8%

          \[\leadsto {\left({\left({\color{blue}{\left(a \cdot \left(-\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       7. associate-*r* 71.8%

          \[\leadsto {\left({\left({\left(a \cdot \left(-\sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       8. *-commutative 71.8%

          \[\leadsto {\left({\left({\left(a \cdot \left(-\sin \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.005555555555555556\right)\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       9. *-commutative 71.8%

          \[\leadsto {\left({\left({\left(a \cdot \left(-\sin \color{blue}{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

       10. *-commutative 71.8%

           \[\leadsto {\left({\left({\left(a \cdot \left(-\sin \left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

   11. Simplified 71.8%

       \[\leadsto {\left({\left({\color{blue}{\left(a \cdot \left(-\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}^{2}\right)}^{3}\right)}^{0.3333333333333333} \]

   13. Applied egg-rr 84.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.1e+148)
   (* b b)
   (pow (* a (sin (* 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 <= 1.1e+148) {
		tmp = b * b;
	} else {
		tmp = pow((a * sin((((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 <= 1.1e+148) {
		tmp = b * b;
	} else {
		tmp = Math.pow((a * Math.sin((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 <= 1.1e+148:
		tmp = b * b
	else:
		tmp = math.pow((a * math.sin((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 <= 1.1e+148)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * sin(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 <= 1.1e+148)
		tmp = b * b;
	else
		tmp = (a * sin((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, 1.1e+148], N[(b * b), $MachinePrecision], N[Power[N[(a * N[Sin[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 < 1.0999999999999999e148

   1. Initial program 78.5%

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

      1. associate-*l/ 78.5%

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

      2. associate-/l* 78.4%

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

      3. cos-neg 78.4%

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

      4. distribute-lft-neg-out 78.4%

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

      5. distribute-frac-neg 78.4%

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

      6. distribute-frac-neg 78.4%

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

      7. distribute-lft-neg-out 78.4%

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

      8. cos-neg 78.4%

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

      9. associate-*l/ 78.4%

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

      10. associate-/l* 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in angle around 0 63.1%

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

      1. unpow2 63.1%

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

   7. Applied egg-rr 63.1%

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

   if 1.0999999999999999e148 < a

   1. Initial program 97.4%

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

      1. associate-*l/ 97.3%

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

      2. associate-/l* 97.5%

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

      3. cos-neg 97.5%

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

      4. distribute-lft-neg-out 97.5%

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

      5. distribute-frac-neg 97.5%

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

      6. distribute-frac-neg 97.5%

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

      7. distribute-lft-neg-out 97.5%

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

      8. cos-neg 97.5%

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

      9. associate-*l/ 97.5%

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

      10. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

   6. Applied egg-rr 66.8%

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

   8. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{{\left({\left({\left(\mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot b\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. Taylor expanded in a around inf 66.7%

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

       1. *-commutative 66.7%

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

       2. associate-*r* 66.8%

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

       3. *-commutative 66.8%

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

       4. associate-*l* 66.9%

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

       5. *-commutative 66.9%

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

       6. unpow2 66.9%

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

       7. unpow2 66.9%

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

       8. swap-sqr 84.2%

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

       9. unpow2 84.2%

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

       10. *-commutative 84.2%

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

       11. *-commutative 84.2%

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

       12. associate-*l* 84.1%

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

   11. Simplified 84.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.1 \cdot 10^{+148}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.5% accurate, 139.0× speedup

Math

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

FPCore

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

C

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

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 * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. associate-*l/ 81.0%

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

   2. associate-/l* 81.0%

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

   3. cos-neg 81.0%

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

   4. distribute-lft-neg-out 81.0%

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

   5. distribute-frac-neg 81.0%

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

   6. distribute-frac-neg 81.0%

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

   7. distribute-lft-neg-out 81.0%

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

   8. cos-neg 81.0%

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

   9. associate-*l/ 80.9%

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

   10. associate-/l* 80.9%

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

3. Simplified 80.9%

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

5. Taylor expanded in angle around 0 59.3%

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

   1. unpow2 59.3%

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

7. Applied egg-rr 59.3%

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

Reproduce

herbie shell --seed 2024123 
(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)))