ab-angle->ABCF A

Percentage Accurate: 79.3% → 79.3%

Time: 20.7s

Alternatives: 14

Speedup: 1.0×

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Exp[N[Log[1 + N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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 66.8%

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

   2. associate-*l/ 66.8%

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

   3. associate-*r/ 66.8%

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

   4. div-inv 66.8%

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

   5. metadata-eval 66.8%

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

4. Applied egg-rr 66.8%

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

Alternative 2: 79.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Cos[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[b, 2.0], $MachinePrecision]), $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/ 81.0%

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

   2. clear-num 81.0%

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

4. Applied egg-rr 81.0%

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

   1. add-exp-log 44.1%

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

   2. associate-/r* 44.1%

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

6. Applied egg-rr 44.1%

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

   1. rem-exp-log 44.1%

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

   2. *-un-lft-identity 44.1%

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

   3. exp-prod 44.1%

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

   4. rem-exp-log 44.1%

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

   5. associate-/r* 44.1%

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

8. Applied egg-rr 44.1%

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

   1. *-commutative 44.1%

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

   2. unpow-prod-down 44.1%

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

   3. pow-exp 44.1%

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

   4. *-un-lft-identity 44.1%

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

   5. add-exp-log 81.0%

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

10. Applied egg-rr 81.0%

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

Alternative 3: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $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. associate-*l/ 81.0%

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

   2. clear-num 81.0%

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

4. Applied egg-rr 81.0%

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

Alternative 4: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)))
   (+ (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 / 180.0) * ((double) M_PI);
	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 / 180.0) * Math.PI;
	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 / 180.0) * math.pi
	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(Float64(angle_m / 180.0) * pi)
	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 / 180.0) * pi;
	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[(N[(angle$95$m / 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]]

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. Add Preprocessing

Alternative 5: 79.4% 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. unpow2 81.0%

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

   2. associate-*l/ 80.9%

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

   3. associate-/l* 81.0%

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

   4. unpow2 81.0%

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

3. Simplified 81.0%

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

Alternative 6: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[Power[N[Sqrt[N[(a * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Cos[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 66.8%

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

   2. associate-*l/ 66.8%

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

   3. associate-*r/ 66.8%

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

   4. div-inv 66.8%

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

   5. metadata-eval 66.8%

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

4. Applied egg-rr 66.8%

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

   1. add-sqr-sqrt 66.8%

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

   2. pow2 66.8%

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

6. Applied egg-rr 80.9%

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

Alternative 7: 79.3% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle_m PI))))
   (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 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	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 = 0.005555555555555556 * (angle_m * Math.PI);
	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 = 0.005555555555555556 * (angle_m * math.pi)
	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(0.005555555555555556 * Float64(angle_m * pi))
	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 = 0.005555555555555556 * (angle_m * pi);
	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[(0.005555555555555556 * N[(angle$95$m * Pi), $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. unpow2 81.0%

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

   2. associate-*l/ 80.9%

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

   3. associate-/l* 81.0%

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

   4. unpow2 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in a around 0 74.4%

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

   1. *-commutative 74.4%

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

   2. associate-*r* 74.5%

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

   3. unpow2 74.5%

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

   4. unpow2 74.5%

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

   5. unpow2 74.5%

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

   6. unpow2 74.5%

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

   7. swap-sqr 81.0%

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

   8. *-commutative 81.0%

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

7. Simplified 80.9%

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

Alternative 8: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $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. expm1-log1p-u 66.8%

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

   2. associate-*l/ 66.8%

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

   3. associate-*r/ 66.8%

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

   4. div-inv 66.8%

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

   5. metadata-eval 66.8%

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

4. Applied egg-rr 66.8%

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

5. Taylor expanded in angle around 0 80.8%

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

Alternative 9: 58.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := a \cdot t\_0\\ \mathbf{if}\;b \leq 1.14 \cdot 10^{-138}:\\ \;\;\;\;{\left(a \cdot \sin t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(angle\_m \cdot \frac{\pi}{180}\right)\right)}^{2} + t\_1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI 0.005555555555555556))) (t_1 (* a t_0)))
   (if (<= b 1.14e-138)
     (pow (* a (sin t_0)) 2.0)
     (+ (pow (* b (cos (* angle_m (/ PI 180.0)))) 2.0) (* t_1 t_1)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * 0.005555555555555556);
	double t_1 = a * t_0;
	double tmp;
	if (b <= 1.14e-138) {
		tmp = pow((a * sin(t_0)), 2.0);
	} else {
		tmp = pow((b * cos((angle_m * (((double) M_PI) / 180.0)))), 2.0) + (t_1 * t_1);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * 0.005555555555555556);
	double t_1 = a * t_0;
	double tmp;
	if (b <= 1.14e-138) {
		tmp = Math.pow((a * Math.sin(t_0)), 2.0);
	} else {
		tmp = Math.pow((b * Math.cos((angle_m * (Math.PI / 180.0)))), 2.0) + (t_1 * t_1);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = angle_m * (math.pi * 0.005555555555555556)
	t_1 = a * t_0
	tmp = 0
	if b <= 1.14e-138:
		tmp = math.pow((a * math.sin(t_0)), 2.0)
	else:
		tmp = math.pow((b * math.cos((angle_m * (math.pi / 180.0)))), 2.0) + (t_1 * t_1)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * 0.005555555555555556))
	t_1 = Float64(a * t_0)
	tmp = 0.0
	if (b <= 1.14e-138)
		tmp = Float64(a * sin(t_0)) ^ 2.0;
	else
		tmp = Float64((Float64(b * cos(Float64(angle_m * Float64(pi / 180.0)))) ^ 2.0) + Float64(t_1 * t_1));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = angle_m * (pi * 0.005555555555555556);
	t_1 = a * t_0;
	tmp = 0.0;
	if (b <= 1.14e-138)
		tmp = (a * sin(t_0)) ^ 2.0;
	else
		tmp = ((b * cos((angle_m * (pi / 180.0)))) ^ 2.0) + (t_1 * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * t$95$0), $MachinePrecision]}, If[LessEqual[b, 1.14e-138], N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(b * N[Cos[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1399999999999999e-138

   1. Initial program 81.6%

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

      1. associate-*l/ 81.6%

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

      2. clear-num 81.6%

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

   4. Applied egg-rr 81.6%

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

      1. add-exp-log 41.5%

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

      2. associate-/r* 41.5%

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

   6. Applied egg-rr 41.5%

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

   7. Taylor expanded in a around inf 49.6%

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

      1. *-commutative 49.6%

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

      2. *-commutative 49.6%

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

      3. associate-*r* 49.7%

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

      4. unpow2 49.7%

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

      5. unpow2 49.7%

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

      6. swap-sqr 52.6%

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

      7. unpow2 52.6%

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

      8. *-commutative 52.6%

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

   9. Simplified 52.6%

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

   if 1.1399999999999999e-138 < b

   1. Initial program 79.9%

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

      1. unpow2 79.9%

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

      2. associate-*l/ 80.0%

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

      3. associate-/l* 80.0%

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

      4. unpow2 80.0%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in angle around 0 66.0%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]
   6. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

      2. associate-*l* 66.0%

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

      3. unpow2 66.0%

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

      4. unpow2 66.0%

         \[\leadsto {a}^{2} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

      5. unswap-sqr 66.0%

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

      6. metadata-eval 66.0%

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

      7. swap-sqr 65.9%

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

      8. associate-*r* 66.0%

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

      9. associate-*r* 65.9%

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

      10. unpow2 65.9%

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

      11. associate-*r* 65.9%

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

      12. *-commutative 65.9%

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

   7. Simplified 65.9%

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

      1. add-sqr-sqrt 65.9%

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

      2. sqrt-unprod 66.0%

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

      3. pow-prod-down 66.0%

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

      4. pow-prod-down 75.8%

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

      5. pow-prod-up 75.8%

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

      6. *-commutative 75.8%

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

      7. associate-*l* 75.8%

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

      8. metadata-eval 75.8%

         \[\leadsto \sqrt{{\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{\color{blue}{4}}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

   9. Applied egg-rr 75.8%

      \[\leadsto \color{blue}{\sqrt{{\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{4}}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]
   10. Step-by-step derivation

       1. associate-*r* 75.8%

          \[\leadsto \sqrt{{\color{blue}{\left(\left(a \cdot angle\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{4}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

   11. Simplified 75.8%

       \[\leadsto \color{blue}{\sqrt{{\left(\left(a \cdot angle\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]
   12. Step-by-step derivation

       1. sqrt-pow1 77.9%

          \[\leadsto \color{blue}{{\left(\left(a \cdot angle\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{\left(\frac{4}{2}\right)}} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} \]

       2. metadata-eval 77.9%

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

       3. pow2 77.9%

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

       4. associate-*l* 78.0%

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

       5. associate-*l* 77.9%

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

   13. Applied egg-rr 77.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.14 \cdot 10^{-138}:\\ \;\;\;\;{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ \mathbf{if}\;a \leq 9.6 \cdot 10^{+159}:\\ \;\;\;\;{\left(b \cdot \cos t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin t\_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle_m PI))))
   (if (<= a 9.6e+159) (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 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	double tmp;
	if (a <= 9.6e+159) {
		tmp = pow((b * cos(t_0)), 2.0);
	} else {
		tmp = pow((a * sin(t_0)), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (angle_m * Math.PI);
	double tmp;
	if (a <= 9.6e+159) {
		tmp = Math.pow((b * Math.cos(t_0)), 2.0);
	} else {
		tmp = Math.pow((a * Math.sin(t_0)), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = 0.005555555555555556 * (angle_m * math.pi)
	tmp = 0
	if a <= 9.6e+159:
		tmp = math.pow((b * math.cos(t_0)), 2.0)
	else:
		tmp = math.pow((a * math.sin(t_0)), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle_m * pi))
	tmp = 0.0
	if (a <= 9.6e+159)
		tmp = Float64(b * cos(t_0)) ^ 2.0;
	else
		tmp = Float64(a * sin(t_0)) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = 0.005555555555555556 * (angle_m * pi);
	tmp = 0.0;
	if (a <= 9.6e+159)
		tmp = (b * cos(t_0)) ^ 2.0;
	else
		tmp = (a * sin(t_0)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 9.6e+159], 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]]]

Derivation

1. Split input into 2 regimes

2. if a < 9.5999999999999999e159

   1. Initial program 79.2%

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

      1. unpow2 79.2%

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

      2. associate-*l/ 79.1%

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

      3. associate-/l* 79.2%

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

      4. unpow2 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in a around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. *-commutative 60.9%

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

      3. associate-*r* 60.9%

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

      4. unpow2 60.9%

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

      5. unpow2 60.9%

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

      6. swap-sqr 60.9%

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

      7. unpow2 60.9%

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

      8. *-commutative 60.9%

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

      9. associate-*r* 60.9%

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

      10. *-commutative 60.9%

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

   7. Simplified 60.9%

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

   if 9.5999999999999999e159 < a

   1. Initial program 96.2%

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

      1. associate-*l/ 96.2%

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

      2. clear-num 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. add-cube-cbrt 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}} \cdot \sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}}\right)}\right)}^{2} \]

      2. pow3 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}}\right)}^{3}\right)}\right)}^{2} \]

      3. associate-/r/ 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\frac{1}{180} \cdot \left(angle \cdot \pi\right)}}\right)}^{3}\right)\right)}^{2} \]

      4. metadata-eval 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt[3]{\color{blue}{0.005555555555555556} \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)\right)}^{2} \]

      5. *-commutative 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt[3]{0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

   6. Applied egg-rr 96.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left(\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}\right)}^{3}\right)}\right)}^{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\color{blue}{\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}} \cdot \sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}}^{3}\right)\right)}^{2} \]

      2. unpow-prod-down 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)}\right)}^{2} \]

      3. pow1/3 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt{\color{blue}{{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{0.3333333333333333}}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      4. *-commutative 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt{{\left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)}^{0.3333333333333333}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      5. sqrt-pow1 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\color{blue}{\left({\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      6. *-commutative 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      7. associate-*l* 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      8. metadata-eval 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      9. pow1/3 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{\color{blue}{{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{0.3333333333333333}}}\right)}^{3}\right)\right)}^{2} \]

      10. *-commutative 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{{\left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)}^{0.3333333333333333}}\right)}^{3}\right)\right)}^{2} \]

      11. sqrt-pow1 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left({\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}\right)\right)}^{2} \]

      12. *-commutative 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)\right)}^{2} \]

      13. associate-*l* 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)\right)}^{2} \]

      14. metadata-eval 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}\right)\right)}^{2} \]

   8. Applied egg-rr 55.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3}\right)}\right)}^{2} \]
   9. Step-by-step derivation

      1. pow-sqr 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{\left(2 \cdot 3\right)}\right)}\right)}^{2} \]

      2. metadata-eval 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{\color{blue}{6}}\right)\right)}^{2} \]

   10. Simplified 55.4%

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{6}\right)}\right)}^{2} \]

   11. Taylor expanded in a around inf 70.8%

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

       1. unpow2 70.8%

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

       2. *-commutative 70.8%

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

       3. associate-*r* 70.8%

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

       4. unpow2 70.8%

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

       5. swap-sqr 85.5%

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

       6. unpow2 85.5%

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

       7. associate-*r* 85.5%

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

       8. *-commutative 85.5%

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

   13. Simplified 85.5%

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

Alternative 11: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.6 \cdot 10^{+159}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.6e+159)
   (* b b)
   (pow (* a (sin (* 0.005555555555555556 (* angle_m PI)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.6e+159) {
		tmp = b * b;
	} else {
		tmp = pow((a * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))), 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 <= 9.6e+159) {
		tmp = b * b;
	} else {
		tmp = Math.pow((a * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 9.6e+159:
		tmp = b * b
	else:
		tmp = math.pow((a * math.sin((0.005555555555555556 * (angle_m * math.pi)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 9.6e+159)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 9.6e+159)
		tmp = b * b;
	else
		tmp = (a * sin((0.005555555555555556 * (angle_m * pi)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 9.6e+159], N[(b * b), $MachinePrecision], N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.5999999999999999e159

   1. Initial program 79.2%

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

      1. unpow2 79.2%

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

      2. associate-*l/ 79.1%

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

      3. associate-/l* 79.2%

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

      4. unpow2 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in angle around 0 60.7%

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

      1. unpow2 60.7%

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

   7. Applied egg-rr 60.7%

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

   if 9.5999999999999999e159 < a

   1. Initial program 96.2%

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

      1. associate-*l/ 96.2%

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

      2. clear-num 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. add-cube-cbrt 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}} \cdot \sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}}\right)}\right)}^{2} \]

      2. pow3 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\frac{1}{\frac{180}{angle \cdot \pi}}}\right)}^{3}\right)}\right)}^{2} \]

      3. associate-/r/ 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\frac{1}{180} \cdot \left(angle \cdot \pi\right)}}\right)}^{3}\right)\right)}^{2} \]

      4. metadata-eval 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt[3]{\color{blue}{0.005555555555555556} \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)\right)}^{2} \]

      5. *-commutative 96.2%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt[3]{0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

   6. Applied egg-rr 96.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left(\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}\right)}^{3}\right)}\right)}^{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\color{blue}{\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}} \cdot \sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}}^{3}\right)\right)}^{2} \]

      2. unpow-prod-down 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)}\right)}^{2} \]

      3. pow1/3 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt{\color{blue}{{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{0.3333333333333333}}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      4. *-commutative 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left(\sqrt{{\left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)}^{0.3333333333333333}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      5. sqrt-pow1 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\color{blue}{\left({\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      6. *-commutative 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      7. associate-*l* 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      8. metadata-eval 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \cdot {\left(\sqrt{\sqrt[3]{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}}\right)}^{3}\right)\right)}^{2} \]

      9. pow1/3 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{\color{blue}{{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{0.3333333333333333}}}\right)}^{3}\right)\right)}^{2} \]

      10. *-commutative 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt{{\left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)}^{0.3333333333333333}}\right)}^{3}\right)\right)}^{2} \]

      11. sqrt-pow1 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left({\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{3}\right)\right)}^{2} \]

      12. *-commutative 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)\right)}^{2} \]

      13. associate-*l* 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{3}\right)\right)}^{2} \]

      14. metadata-eval 55.4%

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}\right)\right)}^{2} \]

   8. Applied egg-rr 55.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{3}\right)}\right)}^{2} \]
   9. Step-by-step derivation

      1. pow-sqr 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{\left(2 \cdot 3\right)}\right)}\right)}^{2} \]

      2. metadata-eval 55.4%

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{\color{blue}{6}}\right)\right)}^{2} \]

   10. Simplified 55.4%

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left({\left({\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{0.16666666666666666}\right)}^{6}\right)}\right)}^{2} \]

   11. Taylor expanded in a around inf 70.8%

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

       1. unpow2 70.8%

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

       2. *-commutative 70.8%

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

       3. associate-*r* 70.8%

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

       4. unpow2 70.8%

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

       5. swap-sqr 85.5%

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

       6. unpow2 85.5%

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

       7. associate-*r* 85.5%

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

       8. *-commutative 85.5%

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

   13. Simplified 85.5%

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

Alternative 12: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+134}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{b}^{4}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 7.5e+134) (* b b) (sqrt (pow b 4.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 7.5e+134) {
		tmp = b * b;
	} else {
		tmp = sqrt(pow(b, 4.0));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= 7.5d+134) then
        tmp = b * b
    else
        tmp = sqrt((b ** 4.0d0))
    end if
    code = tmp
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 7.5e+134) {
		tmp = b * b;
	} else {
		tmp = Math.sqrt(Math.pow(b, 4.0));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 7.5e+134:
		tmp = b * b
	else:
		tmp = math.sqrt(math.pow(b, 4.0))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 7.5e+134)
		tmp = Float64(b * b);
	else
		tmp = sqrt((b ^ 4.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 7.5e+134)
		tmp = b * b;
	else
		tmp = sqrt((b ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 7.5e+134], N[(b * b), $MachinePrecision], N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.5000000000000001e134

   1. Initial program 80.2%

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

      1. unpow2 80.2%

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

      2. associate-*l/ 80.0%

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

      3. associate-/l* 80.1%

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

      4. unpow2 80.1%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in angle around 0 61.8%

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

      1. unpow2 61.8%

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

   7. Applied egg-rr 61.8%

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

   if 7.5000000000000001e134 < a

   1. Initial program 86.9%

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

      1. unpow2 86.9%

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

      2. associate-*l/ 86.9%

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

      3. associate-/l* 87.1%

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

      4. unpow2 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in angle around 0 29.5%

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

      1. unpow2 29.5%

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

   7. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{b \cdot b} \]
   8. Step-by-step derivation

      1. pow2 29.5%

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

      2. add-sqr-sqrt 29.5%

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

      3. sqrt-unprod 41.0%

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

      4. pow-prod-up 41.0%

         \[\leadsto \sqrt{\color{blue}{{b}^{\left(2 + 2\right)}}} \]

      5. metadata-eval 41.0%

         \[\leadsto \sqrt{{b}^{\color{blue}{4}}} \]

   9. Applied egg-rr 41.0%

      \[\leadsto \color{blue}{\sqrt{{b}^{4}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.3 \cdot 10^{+231}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{b}^{6}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 4.3e+231) (* b b) (cbrt (pow b 6.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.3e+231) {
		tmp = b * b;
	} else {
		tmp = cbrt(pow(b, 6.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.3e+231) {
		tmp = b * b;
	} else {
		tmp = Math.cbrt(Math.pow(b, 6.0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.3e+231)
		tmp = Float64(b * b);
	else
		tmp = cbrt((b ^ 6.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 4.3e+231], N[(b * b), $MachinePrecision], N[Power[N[Power[b, 6.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.29999999999999976e231

   1. Initial program 82.3%

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

      1. unpow2 82.3%

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

      2. associate-*l/ 82.2%

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

      3. associate-/l* 82.3%

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

      4. unpow2 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in angle around 0 58.9%

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

      1. unpow2 58.9%

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

   7. Applied egg-rr 58.9%

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

   if 4.29999999999999976e231 < angle

   1. Initial program 67.8%

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

      1. unpow2 67.8%

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

      2. associate-*l/ 67.2%

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

      3. associate-/l* 67.4%

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

      4. unpow2 67.4%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in angle around 0 44.3%

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

      1. unpow2 44.3%

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

   7. Applied egg-rr 44.3%

      \[\leadsto \color{blue}{b \cdot b} \]
   8. Step-by-step derivation

      1. pow2 44.3%

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

      2. add-sqr-sqrt 44.3%

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

      3. sqrt-unprod 56.8%

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

      4. pow-prod-up 56.8%

         \[\leadsto \sqrt{\color{blue}{{b}^{\left(2 + 2\right)}}} \]

      5. metadata-eval 56.8%

         \[\leadsto \sqrt{{b}^{\color{blue}{4}}} \]

   9. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{\sqrt{{b}^{4}}} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 56.3%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{{b}^{4}} \cdot \sqrt{{b}^{4}}\right) \cdot \sqrt{{b}^{4}}}} \]

       2. pow1/3 56.3%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{{b}^{4}} \cdot \sqrt{{b}^{4}}\right) \cdot \sqrt{{b}^{4}}\right)}^{0.3333333333333333}} \]

       3. add-sqr-sqrt 56.3%

          \[\leadsto {\left(\color{blue}{{b}^{4}} \cdot \sqrt{{b}^{4}}\right)}^{0.3333333333333333} \]

       4. sqrt-pow1 56.3%

          \[\leadsto {\left({b}^{4} \cdot \color{blue}{{b}^{\left(\frac{4}{2}\right)}}\right)}^{0.3333333333333333} \]

       5. metadata-eval 56.3%

          \[\leadsto {\left({b}^{4} \cdot {b}^{\color{blue}{2}}\right)}^{0.3333333333333333} \]

       6. pow-prod-up 56.3%

          \[\leadsto {\color{blue}{\left({b}^{\left(4 + 2\right)}\right)}}^{0.3333333333333333} \]

       7. metadata-eval 56.3%

          \[\leadsto {\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 56.3%

       \[\leadsto \color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 56.3%

          \[\leadsto \color{blue}{\sqrt[3]{{b}^{6}}} \]

   13. Simplified 56.3%

       \[\leadsto \color{blue}{\sqrt[3]{{b}^{6}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.4% 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. unpow2 81.0%

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

   2. associate-*l/ 80.9%

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

   3. associate-/l* 81.0%

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

   4. unpow2 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in angle around 0 57.6%

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

   1. unpow2 57.6%

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

7. Applied egg-rr 57.6%

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

Reproduce

herbie shell --seed 2024131 
(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)))