ab-angle->ABCF B

Percentage Accurate: 54.0% → 64.7%

Time: 46.7s

Alternatives: 27

Speedup: 32.2×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ t_1 := \sqrt[3]{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \left(2 \cdot \pi\right)\right)\right)}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(a, t\_0 \cdot \left(-a\right), {\left(b \cdot \sqrt{t\_0}\right)}^{2}\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {t\_1}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* 0.005555555555555556 PI))))
        (t_1
         (cbrt
          (*
           (- (pow b 2.0) (pow a 2.0))
           (sin (* angle_m (* 0.005555555555555556 (* 2.0 PI))))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-14)
      (*
       2.0
       (*
        (fma a (* t_0 (- a)) (pow (* b (sqrt t_0)) 2.0))
        (cos (* (/ angle_m 180.0) PI))))
      (* t_1 (pow t_1 2.0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((angle_m * (0.005555555555555556 * ((double) M_PI))));
	double t_1 = cbrt(((pow(b, 2.0) - pow(a, 2.0)) * sin((angle_m * (0.005555555555555556 * (2.0 * ((double) M_PI)))))));
	double tmp;
	if ((angle_m / 180.0) <= 5e-14) {
		tmp = 2.0 * (fma(a, (t_0 * -a), pow((b * sqrt(t_0)), 2.0)) * cos(((angle_m / 180.0) * ((double) M_PI))));
	} else {
		tmp = t_1 * pow(t_1, 2.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	t_1 = cbrt(Float64(Float64((b ^ 2.0) - (a ^ 2.0)) * sin(Float64(angle_m * Float64(0.005555555555555556 * Float64(2.0 * pi))))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-14)
		tmp = Float64(2.0 * Float64(fma(a, Float64(t_0 * Float64(-a)), (Float64(b * sqrt(t_0)) ^ 2.0)) * cos(Float64(Float64(angle_m / 180.0) * pi))));
	else
		tmp = Float64(t_1 * (t_1 ^ 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(0.005555555555555556 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-14], N[(2.0 * N[(N[(a * N[(t$95$0 * (-a)), $MachinePrecision] + N[Power[N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e-14

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. associate-*l* 60.6%

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

   3. Simplified 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. difference-of-squares 63.1%

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

   6. Applied egg-rr 63.1%

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

   7. Taylor expanded in a around 0 70.8%

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

      1. fma-define 71.8%

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

   9. Simplified 72.5%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 51.9%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \color{blue}{\sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       2. pow2 51.9%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \color{blue}{{\left(\sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}}\right)}^{2}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       3. *-commutative 51.9%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left(\sqrt{\color{blue}{{b}^{2} \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       4. sqrt-prod 31.1%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\color{blue}{\left(\sqrt{{b}^{2}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       5. sqrt-pow1 35.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left(\color{blue}{{b}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       6. metadata-eval 35.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left({b}^{\color{blue}{1}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       7. pow1 35.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left(\color{blue}{b} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   11. Applied egg-rr 35.7%

       \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \color{blue}{{\left(b \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   if 5.0000000000000002e-14 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-*l* 31.9%

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

   3. Simplified 31.9%

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

      1. add-cbrt-cube 29.7%

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

      2. pow1/3 22.2%

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

   6. Applied egg-rr 22.2%

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

      1. add-cube-cbrt 22.2%

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

   8. Applied egg-rr 34.4%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(-a\right), {\left(b \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \left(2 \cdot \pi\right)\right)\right)} \cdot {\left(\sqrt[3]{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \left(2 \cdot \pi\right)\right)\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 0.0005:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(a, t\_0 \cdot \left(-a\right), {\left(b \cdot \sqrt{t\_0}\right)}^{2}\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\sqrt[3]{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)}\right)}^{3} \cdot \cos \left(\frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* 0.005555555555555556 PI)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 0.0005)
      (*
       2.0
       (*
        (fma a (* t_0 (- a)) (pow (* b (sqrt t_0)) 2.0))
        (cos (* (/ angle_m 180.0) PI))))
      (*
       2.0
       (*
        (pow
         (cbrt
          (*
           (- (pow b 2.0) (pow a 2.0))
           (sin (* 0.005555555555555556 (* angle_m PI)))))
         3.0)
        (cos (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((angle_m * (0.005555555555555556 * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 0.0005) {
		tmp = 2.0 * (fma(a, (t_0 * -a), pow((b * sqrt(t_0)), 2.0)) * cos(((angle_m / 180.0) * ((double) M_PI))));
	} else {
		tmp = 2.0 * (pow(cbrt(((pow(b, 2.0) - pow(a, 2.0)) * sin((0.005555555555555556 * (angle_m * ((double) M_PI)))))), 3.0) * cos(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 0.0005)
		tmp = Float64(2.0 * Float64(fma(a, Float64(t_0 * Float64(-a)), (Float64(b * sqrt(t_0)) ^ 2.0)) * cos(Float64(Float64(angle_m / 180.0) * pi))));
	else
		tmp = Float64(2.0 * Float64((cbrt(Float64(Float64((b ^ 2.0) - (a ^ 2.0)) * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))) ^ 3.0) * cos(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 0.0005], N[(2.0 * N[(N[(a * N[(t$95$0 * (-a)), $MachinePrecision] + N[Power[N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[Power[N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000001e-4

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. associate-*l* 60.6%

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

   3. Simplified 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. difference-of-squares 63.1%

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

   6. Applied egg-rr 63.1%

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

   7. Taylor expanded in a around 0 70.8%

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

      1. fma-define 71.8%

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

   9. Simplified 72.5%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 51.9%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \color{blue}{\sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       2. pow2 51.9%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \color{blue}{{\left(\sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}}\right)}^{2}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       3. *-commutative 51.9%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left(\sqrt{\color{blue}{{b}^{2} \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       4. sqrt-prod 31.1%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\color{blue}{\left(\sqrt{{b}^{2}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       5. sqrt-pow1 35.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left(\color{blue}{{b}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       6. metadata-eval 35.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left({b}^{\color{blue}{1}} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

       7. pow1 35.7%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), {\left(\color{blue}{b} \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   11. Applied egg-rr 35.7%

       \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \color{blue}{{\left(b \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   if 5.0000000000000001e-4 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. associate-*l* 31.9%

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

   3. Simplified 31.9%

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

      1. unpow2 31.9%

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

      2. unpow2 31.9%

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

      3. difference-of-squares 39.3%

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

   6. Applied egg-rr 39.3%

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

      1. add-cube-cbrt 36.2%

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

      2. pow2 36.2%

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

   8. Applied egg-rr 36.2%

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

      1. add-cube-cbrt 36.2%

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

      2. pow3 36.2%

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

      3. *-commutative 36.2%

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

      4. div-inv 36.0%

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

      5. metadata-eval 36.0%

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

      6. associate-*r* 37.3%

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

      7. *-commutative 37.3%

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

      8. difference-of-squares 29.9%

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

      9. unpow2 29.9%

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

      10. unpow2 29.9%

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

   10. Applied egg-rr 29.9%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 0.0005:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(-a\right), {\left(b \cdot \sqrt{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{2}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\sqrt[3]{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{3} \cdot \cos \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ t_1 := \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(a, t\_0 \cdot \left(-a\right), t\_0 \cdot {b}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(b \cdot \left(b \cdot t\_1\right) - {a}^{2} \cdot t\_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* 0.005555555555555556 PI))))
        (t_1 (sin (* PI (* angle_m 0.005555555555555556)))))
   (*
    angle_s
    (if (<= (- (pow b 2.0) (pow a 2.0)) -2e+101)
      (* 2.0 (fma a (* t_0 (- a)) (* t_0 (pow b 2.0))))
      (*
       2.0
       (*
        (cos (* (/ angle_m 180.0) PI))
        (- (* b (* b t_1)) (* (pow a 2.0) t_1))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((angle_m * (0.005555555555555556 * ((double) M_PI))));
	double t_1 = sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -2e+101) {
		tmp = 2.0 * fma(a, (t_0 * -a), (t_0 * pow(b, 2.0)));
	} else {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * ((b * (b * t_1)) - (pow(a, 2.0) * t_1)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	t_1 = sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -2e+101)
		tmp = Float64(2.0 * fma(a, Float64(t_0 * Float64(-a)), Float64(t_0 * (b ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(b * Float64(b * t_1)) - Float64((a ^ 2.0) * t_1))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -2e+101], N[(2.0 * N[(a * N[(t$95$0 * (-a)), $MachinePrecision] + N[(t$95$0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -2e101

   1. Initial program 50.8%

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

      1. associate-*l* 50.8%

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

      2. associate-*l* 50.8%

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

   3. Simplified 50.8%

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

      1. unpow2 50.8%

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

      2. unpow2 50.8%

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

      3. difference-of-squares 50.8%

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

   6. Applied egg-rr 50.8%

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

   7. Taylor expanded in a around 0 71.5%

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

      1. fma-define 71.5%

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

   9. Simplified 76.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   10. Taylor expanded in angle around 0 73.5%

       \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right) \cdot \color{blue}{1}\right) \]

   if -2e101 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 56.3%

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

      1. associate-*l* 56.3%

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

      2. associate-*l* 56.3%

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

   3. Simplified 56.3%

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

      1. unpow2 56.3%

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

      2. unpow2 56.3%

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

      3. difference-of-squares 61.5%

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

   6. Applied egg-rr 61.5%

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

   7. Taylor expanded in b around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

      3. unsub-neg 66.6%

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

   9. Simplified 65.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(-a\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(b \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) - {a}^{2} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ t_1 := \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+140}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \left(b \cdot \left(t\_1 \cdot b\right) - t\_1 \cdot {a}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \mathsf{fma}\left(t\_0, {b}^{2}, a \cdot \left(a \cdot \left(-t\_0\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* PI (* angle_m 0.005555555555555556))))
        (t_1 (sin (* angle_m (* 0.005555555555555556 PI))))
        (t_2 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= a 1.15e+140)
      (* 2.0 (* t_2 (- (* b (* t_1 b)) (* t_1 (pow a 2.0)))))
      (* 2.0 (* t_2 (fma t_0 (pow b 2.0) (* a (* a (- t_0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	double t_1 = sin((angle_m * (0.005555555555555556 * ((double) M_PI))));
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if (a <= 1.15e+140) {
		tmp = 2.0 * (t_2 * ((b * (t_1 * b)) - (t_1 * pow(a, 2.0))));
	} else {
		tmp = 2.0 * (t_2 * fma(t_0, pow(b, 2.0), (a * (a * -t_0))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))
	t_1 = sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (a <= 1.15e+140)
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(b * Float64(t_1 * b)) - Float64(t_1 * (a ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(t_2 * fma(t_0, (b ^ 2.0), Float64(a * Float64(a * Float64(-t_0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 1.15e+140], N[(2.0 * N[(t$95$2 * N[(N[(b * N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(t$95$0 * N[Power[b, 2.0], $MachinePrecision] + N[(a * N[(a * (-t$95$0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999995e140

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. associate-*l* 55.9%

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

   3. Simplified 55.9%

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

      1. unpow2 55.9%

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

      2. unpow2 55.9%

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

      3. difference-of-squares 58.1%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in b around 0 63.3%

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

      1. +-commutative 63.3%

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

      2. mul-1-neg 63.3%

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

      3. unsub-neg 63.3%

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

   9. Simplified 64.6%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(b \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + b\right)\right) - {a}^{2} \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   if 1.14999999999999995e140 < a

   1. Initial program 45.0%

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

      1. associate-*l* 45.0%

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

      2. associate-*l* 45.0%

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

   3. Simplified 45.0%

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

      1. unpow2 45.0%

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

      2. unpow2 45.0%

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

      3. difference-of-squares 57.5%

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

   6. Applied egg-rr 57.5%

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

   7. Taylor expanded in a around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. *-commutative 71.7%

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

      3. fma-define 71.7%

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

      4. associate-*r* 71.7%

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

      5. *-commutative 71.7%

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

      6. distribute-lft-in 71.7%

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

      7. associate-*r* 71.7%

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

      8. distribute-rgt1-in 71.7%

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

   9. Simplified 71.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{+140}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(b \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot b\right) - \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {a}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), {b}^{2}, a \cdot \left(a \cdot \left(-\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ angle\_s \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \mathsf{fma}\left(a, t\_0 \cdot \left(-a\right), t\_0 \cdot {b}^{2}\right)\right)\right) \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* 0.005555555555555556 PI)))))
   (*
    angle_s
    (*
     2.0
     (*
      (cos (* (/ angle_m 180.0) PI))
      (fma a (* t_0 (- a)) (* t_0 (pow b 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((angle_m * (0.005555555555555556 * ((double) M_PI))));
	return angle_s * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * fma(a, (t_0 * -a), (t_0 * pow(b, 2.0)))));
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	return Float64(angle_s * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * fma(a, Float64(t_0 * Float64(-a)), Float64(t_0 * (b ^ 2.0))))))
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a * N[(t$95$0 * (-a)), $MachinePrecision] + N[(t$95$0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

6. Applied egg-rr 58.0%

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

7. Taylor expanded in a around 0 62.5%

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

   1. fma-define 64.0%

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

9. Simplified 65.6%

   \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

10. Final simplification 65.6%

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

Alternative 6: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(-1 + e^{\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e-20)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (*
     2.0
     (*
      (cos (* (/ angle_m 180.0) PI))
      (*
       (* (+ a b) (- b a))
       (+
        -1.0
        (exp (log1p (sin (* PI (* angle_m 0.005555555555555556))))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-20) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (((a + b) * (b - a)) * (-1.0 + exp(log1p(sin((((double) M_PI) * (angle_m * 0.005555555555555556))))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-20) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (((a + b) * (b - a)) * (-1.0 + Math.exp(Math.log1p(Math.sin((Math.PI * (angle_m * 0.005555555555555556))))))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e-20:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (((a + b) * (b - a)) * (-1.0 + math.exp(math.log1p(math.sin((math.pi * (angle_m * 0.005555555555555556))))))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-20)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(-1.0 + exp(log1p(sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-20], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[Exp[N[Log[1 + N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999989e-20

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*l* 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in angle around 0 58.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. difference-of-squares 63.1%

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

   7. Applied egg-rr 61.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 66.4%

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

   if 1.99999999999999989e-20 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. associate-*l* 31.9%

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

   3. Simplified 31.9%

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

      1. unpow2 31.9%

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

      2. unpow2 31.9%

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

      3. difference-of-squares 39.3%

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

   6. Applied egg-rr 39.3%

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

      1. expm1-log1p-u 39.3%

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

      2. expm1-undefine 39.3%

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

      3. div-inv 39.0%

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

      4. metadata-eval 39.0%

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

   8. Applied egg-rr 39.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(-1 + e^{\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ angle\_s \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(b \cdot \left(t\_0 \cdot b\right) - t\_0 \cdot {a}^{2}\right)\right)\right) \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* 0.005555555555555556 PI)))))
   (*
    angle_s
    (*
     2.0
     (*
      (cos (* (/ angle_m 180.0) PI))
      (- (* b (* t_0 b)) (* t_0 (pow a 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((angle_m * (0.005555555555555556 * ((double) M_PI))));
	return angle_s * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * ((b * (t_0 * b)) - (t_0 * pow(a, 2.0)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((angle_m * (0.005555555555555556 * Math.PI)));
	return angle_s * (2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * ((b * (t_0 * b)) - (t_0 * Math.pow(a, 2.0)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.sin((angle_m * (0.005555555555555556 * math.pi)))
	return angle_s * (2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * ((b * (t_0 * b)) - (t_0 * math.pow(a, 2.0)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	return Float64(angle_s * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(b * Float64(t_0 * b)) - Float64(t_0 * (a ^ 2.0))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	t_0 = sin((angle_m * (0.005555555555555556 * pi)));
	tmp = angle_s * (2.0 * (cos(((angle_m / 180.0) * pi)) * ((b * (t_0 * b)) - (t_0 * (a ^ 2.0)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(b * N[(t$95$0 * b), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

6. Applied egg-rr 58.0%

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

7. Taylor expanded in b around 0 61.8%

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

   1. +-commutative 61.8%

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

   2. mul-1-neg 61.8%

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

   3. unsub-neg 61.8%

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

9. Simplified 63.3%

   \[\leadsto 2 \cdot \left(\color{blue}{\left(b \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + b\right)\right) - {a}^{2} \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

10. Final simplification 63.3%

    \[\leadsto 2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(b \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot b\right) - \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {a}^{2}\right)\right) \]
11. Add Preprocessing

Alternative 8: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot \left(angle\_m \cdot {b}^{2}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-151)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (*
     2.0
     (*
      (cos (* (/ angle_m 180.0) PI))
      (* 0.005555555555555556 (* PI (* angle_m (pow b 2.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-151) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (0.005555555555555556 * (((double) M_PI) * (angle_m * pow(b, 2.0)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-151) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (0.005555555555555556 * (Math.PI * (angle_m * Math.pow(b, 2.0)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-151:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (0.005555555555555556 * (math.pi * (angle_m * math.pow(b, 2.0)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-151)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(0.005555555555555556 * Float64(pi * Float64(angle_m * (b ^ 2.0))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-151)
		tmp = (0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	else
		tmp = 2.0 * (cos(((angle_m / 180.0) * pi)) * (0.005555555555555556 * (pi * (angle_m * (b ^ 2.0)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-151], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(0.005555555555555556 * N[(Pi * N[(angle$95$m * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999994e-152

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

      2. *-commutative 53.0%

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

      3. associate-*l* 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in angle around 0 52.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 53.0%

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

      2. unpow2 53.0%

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

      3. difference-of-squares 53.0%

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

   7. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 67.4%

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

   if -9.9999999999999994e-152 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 55.5%

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

      1. associate-*l* 55.5%

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

      2. associate-*l* 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in b around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. *-commutative 59.9%

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

      3. associate-*r* 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in angle around 0 55.8%

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

      1. associate-*r* 55.8%

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

      2. *-commutative 55.8%

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

   10. Simplified 55.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(t\_0 \cdot b\right) - t\_0 \cdot {a}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(a, t\_0 \cdot \left(-a\right), t\_0 \cdot {b}^{2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* angle_m (* 0.005555555555555556 PI)))))
   (*
    angle_s
    (if (<= a 5e+49)
      (* 2.0 (- (* b (* t_0 b)) (* t_0 (pow a 2.0))))
      (* 2.0 (fma a (* t_0 (- a)) (* t_0 (pow b 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((angle_m * (0.005555555555555556 * ((double) M_PI))));
	double tmp;
	if (a <= 5e+49) {
		tmp = 2.0 * ((b * (t_0 * b)) - (t_0 * pow(a, 2.0)));
	} else {
		tmp = 2.0 * fma(a, (t_0 * -a), (t_0 * pow(b, 2.0)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))
	tmp = 0.0
	if (a <= 5e+49)
		tmp = Float64(2.0 * Float64(Float64(b * Float64(t_0 * b)) - Float64(t_0 * (a ^ 2.0))));
	else
		tmp = Float64(2.0 * fma(a, Float64(t_0 * Float64(-a)), Float64(t_0 * (b ^ 2.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 5e+49], N[(2.0 * N[(N[(b * N[(t$95$0 * b), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(a * N[(t$95$0 * (-a)), $MachinePrecision] + N[(t$95$0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000004e49

   1. Initial program 55.4%

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

      1. associate-*l* 55.4%

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

      2. associate-*l* 55.4%

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

   3. Simplified 55.4%

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

      1. unpow2 55.4%

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

      2. unpow2 55.4%

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

      3. difference-of-squares 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in b around 0 63.8%

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

      1. +-commutative 63.8%

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

      2. mul-1-neg 63.8%

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

      3. unsub-neg 63.8%

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

   9. Simplified 65.2%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(b \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + b\right)\right) - {a}^{2} \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   10. Taylor expanded in angle around 0 62.5%

       \[\leadsto 2 \cdot \left(\left(b \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + b\right)\right) - {a}^{2} \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \color{blue}{1}\right) \]

   if 5.0000000000000004e49 < a

   1. Initial program 50.9%

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

      1. associate-*l* 50.9%

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

      2. associate-*l* 50.9%

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

   3. Simplified 50.9%

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

      1. unpow2 50.9%

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

      2. unpow2 50.9%

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

      3. difference-of-squares 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in a around 0 66.3%

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

      1. fma-define 72.0%

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

   9. Simplified 74.3%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   10. Taylor expanded in angle around 0 61.5%

       \[\leadsto 2 \cdot \left(\mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(0 + \left(-a\right)\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right) \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+49}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot b\right) - \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {a}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(a, \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(-a\right), \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {b}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-151)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (* 2.0 (* (pow b 2.0) (sin (* 0.005555555555555556 (* angle_m PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-151) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 2.0 * (pow(b, 2.0) * sin((0.005555555555555556 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-151) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 2.0 * (Math.pow(b, 2.0) * Math.sin((0.005555555555555556 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-151:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 2.0 * (math.pow(b, 2.0) * math.sin((0.005555555555555556 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-151)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(2.0 * Float64((b ^ 2.0) * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-151)
		tmp = (0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	else
		tmp = 2.0 * ((b ^ 2.0) * sin((0.005555555555555556 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-151], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999994e-152

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

      2. *-commutative 53.0%

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

      3. associate-*l* 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in angle around 0 52.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 53.0%

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

      2. unpow2 53.0%

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

      3. difference-of-squares 53.0%

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

   7. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 67.4%

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

   if -9.9999999999999994e-152 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 55.5%

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

      1. associate-*l* 55.5%

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

      2. associate-*l* 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in b around inf 59.9%

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

   6. Taylor expanded in angle around 0 54.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({b}^{2} \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-151)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (* 2.0 (* (pow b 2.0) (sin (* PI (* angle_m 0.005555555555555556))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-151) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 2.0 * (pow(b, 2.0) * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-151) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 2.0 * (Math.pow(b, 2.0) * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-151:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 2.0 * (math.pow(b, 2.0) * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-151)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(2.0 * Float64((b ^ 2.0) * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-151)
		tmp = (0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	else
		tmp = 2.0 * ((b ^ 2.0) * sin((pi * (angle_m * 0.005555555555555556))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-151], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999994e-152

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

      2. *-commutative 53.0%

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

      3. associate-*l* 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in angle around 0 52.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 53.0%

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

      2. unpow2 53.0%

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

      3. difference-of-squares 53.0%

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

   7. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 67.4%

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

   if -9.9999999999999994e-152 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 55.5%

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

      1. associate-*l* 55.5%

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

      2. associate-*l* 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in b around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. *-commutative 59.9%

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

      3. associate-*r* 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in angle around 0 52.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({b}^{2} \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(\pi \cdot b\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right), \left(angle\_m \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b 2.0) 2e+258)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (fma
     b
     (* (* PI b) (* angle_m 0.011111111111111112))
     (* (* angle_m PI) (* (pow a 2.0) -0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(b, 2.0) <= 2e+258) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = fma(b, ((((double) M_PI) * b) * (angle_m * 0.011111111111111112)), ((angle_m * ((double) M_PI)) * (pow(a, 2.0) * -0.011111111111111112)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if ((b ^ 2.0) <= 2e+258)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = fma(b, Float64(Float64(pi * b) * Float64(angle_m * 0.011111111111111112)), Float64(Float64(angle_m * pi) * Float64((a ^ 2.0) * -0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 2e+258], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] + N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2.00000000000000011e258

   1. Initial program 59.3%

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

      1. associate-*l* 59.3%

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

      2. *-commutative 59.3%

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

      3. associate-*l* 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in angle around 0 56.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 59.3%

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

      2. unpow2 59.3%

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

      3. difference-of-squares 59.3%

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

   7. Applied egg-rr 56.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 64.6%

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

   if 2.00000000000000011e258 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. *-commutative 42.7%

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

      3. associate-*l* 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in angle around 0 38.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 42.7%

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

      2. unpow2 42.7%

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

      3. difference-of-squares 54.9%

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

   7. Applied egg-rr 50.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in b around 0 56.8%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 56.8%

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

      2. fma-define 63.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, 0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right), -0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   10. Simplified 63.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \left(\pi \cdot b\right) \cdot \left(angle \cdot 0.011111111111111112\right), \left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(\pi \cdot b\right) \cdot \left(angle \cdot 0.011111111111111112\right), \left(angle \cdot \pi\right) \cdot \left({a}^{2} \cdot -0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 10^{+215}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \mathsf{fma}\left(a, \pi \cdot \left(-angle\_m \cdot a\right), \pi \cdot \left(angle\_m \cdot {b}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow a 2.0) 1e+215)
    (+
     (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
     (*
      b
      (+
       (* 0.011111111111111112 (* angle_m (* PI b)))
       (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
    (*
     0.011111111111111112
     (fma a (* PI (- (* angle_m a))) (* PI (* angle_m (pow b 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(a, 2.0) <= 1e+215) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else {
		tmp = 0.011111111111111112 * fma(a, (((double) M_PI) * -(angle_m * a)), (((double) M_PI) * (angle_m * pow(b, 2.0))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if ((a ^ 2.0) <= 1e+215)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	else
		tmp = Float64(0.011111111111111112 * fma(a, Float64(pi * Float64(-Float64(angle_m * a))), Float64(pi * Float64(angle_m * (b ^ 2.0)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 1e+215], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(Pi * (-N[(angle$95$m * a), $MachinePrecision])), $MachinePrecision] + N[(Pi * N[(angle$95$m * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 9.99999999999999907e214

   1. Initial program 60.1%

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

      1. associate-*l* 60.1%

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

      2. *-commutative 60.1%

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

      3. associate-*l* 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in angle around 0 56.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 60.1%

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

      2. unpow2 60.1%

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

      3. difference-of-squares 60.1%

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

   7. Applied egg-rr 56.3%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in b around 0 63.8%

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

   if 9.99999999999999907e214 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 43.7%

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

      1. associate-*l* 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-*l* 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in angle around 0 42.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 43.7%

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

      2. unpow2 43.7%

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

      3. difference-of-squares 54.1%

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

   7. Applied egg-rr 52.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 58.9%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(-1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right)\right) + angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. fma-define 61.2%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + angle \cdot \left(\pi \cdot \left(b + -1 \cdot b\right)\right), angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]

   10. Simplified 61.2%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\mathsf{fma}\left(a, \pi \cdot \left(angle \cdot \left(-a\right)\right), \pi \cdot \left(angle \cdot {b}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 10^{+215}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \mathsf{fma}\left(a, \pi \cdot \left(-angle \cdot a\right), \pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e-105)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (*
     2.0
     (*
      (cos (* (/ angle_m 180.0) PI))
      (*
       (sin (* angle_m (* 0.005555555555555556 PI)))
       (* (+ a b) (- b a))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-105) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (sin((angle_m * (0.005555555555555556 * ((double) M_PI)))) * ((a + b) * (b - a))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-105) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (Math.sin((angle_m * (0.005555555555555556 * Math.PI))) * ((a + b) * (b - a))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-105:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (math.sin((angle_m * (0.005555555555555556 * math.pi))) * ((a + b) * (b - a))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-105)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(sin(Float64(angle_m * Float64(0.005555555555555556 * pi))) * Float64(Float64(a + b) * Float64(b - a)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-105)
		tmp = (0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	else
		tmp = 2.0 * (cos(((angle_m / 180.0) * pi)) * (sin((angle_m * (0.005555555555555556 * pi))) * ((a + b) * (b - a))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-105], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999963e-105

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-*l* 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in angle around 0 56.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. difference-of-squares 60.1%

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

   7. Applied egg-rr 58.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 64.2%

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

   if 4.99999999999999963e-105 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 45.2%

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

      1. associate-*l* 45.2%

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

      2. associate-*l* 45.2%

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

   3. Simplified 45.2%

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

      1. unpow2 45.2%

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

      2. unpow2 45.2%

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

      3. difference-of-squares 52.5%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in angle around inf 52.0%

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

      1. *-commutative 52.0%

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

      2. associate-*r* 55.5%

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

   9. Simplified 55.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e-105)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (*
     2.0
     (*
      (cos (* (/ angle_m 180.0) PI))
      (* (* (+ a b) (- b a)) (sin (* angle_m (/ PI 180.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-105) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (((a + b) * (b - a)) * sin((angle_m * (((double) M_PI) / 180.0)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-105) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (((a + b) * (b - a)) * Math.sin((angle_m * (Math.PI / 180.0)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-105:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (((a + b) * (b - a)) * math.sin((angle_m * (math.pi / 180.0)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-105)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(Float64(a + b) * Float64(b - a)) * sin(Float64(angle_m * Float64(pi / 180.0))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-105)
		tmp = (0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	else
		tmp = 2.0 * (cos(((angle_m / 180.0) * pi)) * (((a + b) * (b - a)) * sin((angle_m * (pi / 180.0)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-105], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999963e-105

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-*l* 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in angle around 0 56.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. difference-of-squares 60.1%

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

   7. Applied egg-rr 58.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 64.2%

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

   if 4.99999999999999963e-105 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 45.2%

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

      1. associate-*l* 45.2%

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

      2. associate-*l* 45.2%

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

   3. Simplified 45.2%

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

      1. unpow2 45.2%

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

      2. unpow2 45.2%

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

      3. difference-of-squares 52.5%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in angle around inf 52.0%

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

      1. *-commutative 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-*r* 52.2%

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

   9. Simplified 52.2%

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

       1. metadata-eval 52.2%

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

       2. div-inv 52.5%

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

       3. clear-num 52.5%

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

       4. un-div-inv 53.7%

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

   11. Applied egg-rr 53.7%

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

       1. associate-/r/ 55.5%

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

       2. associate-*l/ 53.7%

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

       3. *-commutative 53.7%

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

       4. associate-/l* 55.5%

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

   13. Simplified 55.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e-105)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (*
     2.0
     (*
      (* (sin (* PI (* angle_m 0.005555555555555556))) (* (+ a b) (- b a)))
      (cos (* 0.005555555555555556 (* angle_m PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-105) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 2.0 * ((sin((((double) M_PI) * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * cos((0.005555555555555556 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-105) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 2.0 * ((Math.sin((Math.PI * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * Math.cos((0.005555555555555556 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-105:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 2.0 * ((math.sin((math.pi * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * math.cos((0.005555555555555556 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-105)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(pi * Float64(angle_m * 0.005555555555555556))) * Float64(Float64(a + b) * Float64(b - a))) * cos(Float64(0.005555555555555556 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-105)
		tmp = (0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	else
		tmp = 2.0 * ((sin((pi * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * cos((0.005555555555555556 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-105], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999963e-105

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-*l* 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in angle around 0 56.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. difference-of-squares 60.1%

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

   7. Applied egg-rr 58.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 64.2%

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

   if 4.99999999999999963e-105 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 45.2%

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

      1. associate-*l* 45.2%

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

      2. associate-*l* 45.2%

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

   3. Simplified 45.2%

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

      1. unpow2 45.2%

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

      2. unpow2 45.2%

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

      3. difference-of-squares 52.5%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in angle around inf 52.0%

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

      1. *-commutative 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-*r* 52.2%

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

   9. Simplified 52.2%

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

   10. Taylor expanded in angle around inf 51.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a + b\right) \cdot {\left({\left(\pi \cdot \left(a + b\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e+133)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (*
     0.011111111111111112
     (*
      angle_m
      (* (+ a b) (pow (pow (* PI (+ a b)) 3.0) 0.3333333333333333)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+133) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a + b) * pow(pow((((double) M_PI) * (a + b)), 3.0), 0.3333333333333333)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+133) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a + b) * Math.pow(Math.pow((Math.PI * (a + b)), 3.0), 0.3333333333333333)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e+133:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((a + b) * math.pow(math.pow((math.pi * (a + b)), 3.0), 0.3333333333333333)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+133)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a + b) * ((Float64(pi * Float64(a + b)) ^ 3.0) ^ 0.3333333333333333))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+133)
		tmp = (0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b))))));
	else
		tmp = 0.011111111111111112 * (angle_m * ((a + b) * (((pi * (a + b)) ^ 3.0) ^ 0.3333333333333333)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+133], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a + b), $MachinePrecision] * N[Power[N[Power[N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e133

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. *-commutative 58.0%

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

      3. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in angle around 0 55.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 58.0%

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

      2. unpow2 58.0%

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

      3. difference-of-squares 61.6%

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

   7. Applied egg-rr 59.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 62.4%

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

   if 2e133 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. *-commutative 28.9%

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

      3. associate-*l* 28.9%

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

   3. Simplified 28.9%

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

   5. Taylor expanded in angle around 0 21.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. difference-of-squares 32.2%

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

   7. Applied egg-rr 21.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* 21.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 21.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 21.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \left(-a\right)\right)}\right) \]

      4. add-sqr-sqrt 10.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}\right)\right) \]

      5. sqrt-unprod 27.1%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right) \]

      6. sqr-neg 27.1%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \sqrt{\color{blue}{a \cdot a}}\right)\right) \]

      7. sqrt-unprod 16.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}\right)\right) \]

      8. add-sqr-sqrt 27.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{a}\right)\right) \]

   9. Applied egg-rr 27.3%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot a\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 30.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   11. Simplified 30.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]
   12. Step-by-step derivation

       1. add-cbrt-cube 30.9%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)}} \cdot \left(b + a\right)\right)\right) \]

       2. pow1/3 27.7%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{{\left(\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)}^{0.3333333333333333}} \cdot \left(b + a\right)\right)\right) \]

       3. pow3 27.7%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({\color{blue}{\left({\left(\pi \cdot \left(b + a\right)\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(b + a\right)\right)\right) \]

       4. +-commutative 27.7%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left({\left({\left(\pi \cdot \color{blue}{\left(a + b\right)}\right)}^{3}\right)}^{0.3333333333333333} \cdot \left(b + a\right)\right)\right) \]

   13. Applied egg-rr 27.7%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{{\left({\left(\pi \cdot \left(a + b\right)\right)}^{3}\right)}^{0.3333333333333333}} \cdot \left(b + a\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot {\left({\left(\pi \cdot \left(a + b\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \cos \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   2.0
   (*
    (* (sin (* PI (* angle_m 0.005555555555555556))) (* (+ a b) (- b a)))
    (cos (* angle_m (* 0.005555555555555556 PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((sin((((double) M_PI) * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * cos((angle_m * (0.005555555555555556 * ((double) M_PI))))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * ((Math.sin((Math.PI * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * Math.cos((angle_m * (0.005555555555555556 * Math.PI)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * ((math.sin((math.pi * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * math.cos((angle_m * (0.005555555555555556 * math.pi)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(Float64(sin(Float64(pi * Float64(angle_m * 0.005555555555555556))) * Float64(Float64(a + b) * Float64(b - a))) * cos(Float64(angle_m * Float64(0.005555555555555556 * pi))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * ((sin((pi * (angle_m * 0.005555555555555556))) * ((a + b) * (b - a))) * cos((angle_m * (0.005555555555555556 * pi)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[(N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

6. Applied egg-rr 58.0%

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

7. Taylor expanded in angle around inf 59.2%

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

   1. *-commutative 59.2%

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

   2. *-commutative 59.2%

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

   3. associate-*r* 59.7%

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

9. Simplified 59.7%

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

10. Taylor expanded in angle around inf 59.0%

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

    1. *-commutative 59.0%

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

    2. associate-*r* 61.0%

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

12. Simplified 61.0%

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

13. Final simplification 61.0%

    \[\leadsto 2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \]
14. Add Preprocessing

Alternative 19: 52.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a + b\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(a + b\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e+133)
    (+
     (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
     (*
      a
      (+
       (* -0.011111111111111112 (* a (* angle_m PI)))
       (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))
    (*
     0.011111111111111112
     (* angle_m (* (+ a b) (expm1 (log1p (* PI (+ a b))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+133) {
		tmp = (0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b))))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a + b) * expm1(log1p((((double) M_PI) * (a + b))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+133) {
		tmp = (0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b))))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((a + b) * Math.expm1(Math.log1p((Math.PI * (a + b))))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e+133:
		tmp = (0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b))))))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((a + b) * math.expm1(math.log1p((math.pi * (a + b))))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+133)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b)))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a + b) * expm1(log1p(Float64(pi * Float64(a + b)))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+133], N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a + b), $MachinePrecision] * N[(Exp[N[Log[1 + N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e133

   1. Initial program 58.0%

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

      1. associate-*l* 58.0%

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

      2. *-commutative 58.0%

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

      3. associate-*l* 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in angle around 0 55.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 58.0%

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

      2. unpow2 58.0%

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

      3. difference-of-squares 61.6%

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

   7. Applied egg-rr 59.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 62.4%

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

   if 2e133 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. *-commutative 28.9%

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

      3. associate-*l* 28.9%

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

   3. Simplified 28.9%

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

   5. Taylor expanded in angle around 0 21.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. difference-of-squares 32.2%

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

   7. Applied egg-rr 21.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* 21.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \]

      2. sub-neg 21.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

      3. distribute-lft-in 21.0%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \left(-a\right)\right)}\right) \]

      4. add-sqr-sqrt 10.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}\right)\right) \]

      5. sqrt-unprod 27.1%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right) \]

      6. sqr-neg 27.1%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \sqrt{\color{blue}{a \cdot a}}\right)\right) \]

      7. sqrt-unprod 16.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}\right)\right) \]

      8. add-sqr-sqrt 27.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{a}\right)\right) \]

   9. Applied egg-rr 27.3%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot a\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 30.5%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

   11. Simplified 30.5%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]
   12. Step-by-step derivation

       1. expm1-log1p-u 17.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(b + a\right)\right)\right)} \cdot \left(b + a\right)\right)\right) \]

       2. expm1-undefine 18.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(b + a\right)\right)} - 1\right)} \cdot \left(b + a\right)\right)\right) \]

       3. +-commutative 18.0%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(e^{\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(a + b\right)}\right)} - 1\right) \cdot \left(b + a\right)\right)\right) \]

   13. Applied egg-rr 18.0%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(a + b\right)\right)} - 1\right)} \cdot \left(b + a\right)\right)\right) \]
   14. Step-by-step derivation

       1. expm1-define 17.2%

          \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(a + b\right)\right)\right)} \cdot \left(b + a\right)\right)\right) \]

   15. Simplified 17.2%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(a + b\right)\right)\right)} \cdot \left(b + a\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(a + b\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-166}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right) + a \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right) - a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e-166)
    (*
     0.011111111111111112
     (+
      (* angle_m (* PI (pow b 2.0)))
      (* a (- (* angle_m (* PI (- b b))) (* a (* angle_m PI))))))
    (+
     (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
     (*
      b
      (+
       (* 0.011111111111111112 (* angle_m (* PI b)))
       (* 0.011111111111111112 (* angle_m (* PI (- a a))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e-166) {
		tmp = 0.011111111111111112 * ((angle_m * (((double) M_PI) * pow(b, 2.0))) + (a * ((angle_m * (((double) M_PI) * (b - b))) - (a * (angle_m * ((double) M_PI))))));
	} else {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e-166) {
		tmp = 0.011111111111111112 * ((angle_m * (Math.PI * Math.pow(b, 2.0))) + (a * ((angle_m * (Math.PI * (b - b))) - (a * (angle_m * Math.PI)))));
	} else {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e-166:
		tmp = 0.011111111111111112 * ((angle_m * (math.pi * math.pow(b, 2.0))) + (a * ((angle_m * (math.pi * (b - b))) - (a * (angle_m * math.pi)))))
	else:
		tmp = (-0.011111111111111112 * (math.pow(a, 2.0) * (angle_m * math.pi))) + (b * ((0.011111111111111112 * (angle_m * (math.pi * b))) + (0.011111111111111112 * (angle_m * (math.pi * (a - a))))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-166)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(pi * (b ^ 2.0))) + Float64(a * Float64(Float64(angle_m * Float64(pi * Float64(b - b))) - Float64(a * Float64(angle_m * pi))))));
	else
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e-166)
		tmp = 0.011111111111111112 * ((angle_m * (pi * (b ^ 2.0))) + (a * ((angle_m * (pi * (b - b))) - (a * (angle_m * pi)))));
	else
		tmp = (-0.011111111111111112 * ((a ^ 2.0) * (angle_m * pi))) + (b * ((0.011111111111111112 * (angle_m * (pi * b))) + (0.011111111111111112 * (angle_m * (pi * (a - a))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-166], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000004e-166

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. *-commutative 57.6%

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

      3. associate-*l* 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in angle around 0 55.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 57.6%

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

      2. unpow2 57.6%

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

      3. difference-of-squares 59.9%

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

   7. Applied egg-rr 58.6%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in a around 0 63.3%

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

   if 1.00000000000000004e-166 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 47.5%

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

      1. associate-*l* 47.5%

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

      2. *-commutative 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in angle around 0 41.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 47.5%

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

      2. unpow2 47.5%

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

      3. difference-of-squares 53.8%

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

   7. Applied egg-rr 47.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in b around 0 44.1%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-166}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right) + a \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right) - a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 57.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) - \pi \cdot \left(angle\_m \cdot {a}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 2e-87)
    (*
     0.011111111111111112
     (- (* b (* angle_m (* PI b))) (* PI (* angle_m (pow a 2.0)))))
    (*
     2.0
     (*
      (cos (* (/ angle_m 180.0) PI))
      (* (* angle_m 0.005555555555555556) (* PI (* (+ a b) (- b a)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-87) {
		tmp = 0.011111111111111112 * ((b * (angle_m * (((double) M_PI) * b))) - (((double) M_PI) * (angle_m * pow(a, 2.0))));
	} else {
		tmp = 2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * ((angle_m * 0.005555555555555556) * (((double) M_PI) * ((a + b) * (b - a)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-87) {
		tmp = 0.011111111111111112 * ((b * (angle_m * (Math.PI * b))) - (Math.PI * (angle_m * Math.pow(a, 2.0))));
	} else {
		tmp = 2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * ((angle_m * 0.005555555555555556) * (Math.PI * ((a + b) * (b - a)))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 2e-87:
		tmp = 0.011111111111111112 * ((b * (angle_m * (math.pi * b))) - (math.pi * (angle_m * math.pow(a, 2.0))))
	else:
		tmp = 2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * ((angle_m * 0.005555555555555556) * (math.pi * ((a + b) * (b - a)))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-87)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * Float64(angle_m * Float64(pi * b))) - Float64(pi * Float64(angle_m * (a ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(angle_m * 0.005555555555555556) * Float64(pi * Float64(Float64(a + b) * Float64(b - a))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e-87)
		tmp = 0.011111111111111112 * ((b * (angle_m * (pi * b))) - (pi * (angle_m * (a ^ 2.0))));
	else
		tmp = 2.0 * (cos(((angle_m / 180.0) * pi)) * ((angle_m * 0.005555555555555556) * (pi * ((a + b) * (b - a)))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-87], N[(0.011111111111111112 * N[(N[(b * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Pi * N[(angle$95$m * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000004e-87

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. *-commutative 58.2%

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

      3. associate-*l* 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in angle around 0 56.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 58.2%

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

      2. unpow2 58.2%

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

      3. difference-of-squares 60.3%

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

   7. Applied egg-rr 59.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   8. Taylor expanded in b around 0 62.7%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 62.7%

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

      2. mul-1-neg 62.7%

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

      3. unsub-neg 62.7%

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

   10. Simplified 62.7%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(\pi \cdot \left(0 + b\right)\right)\right) - \pi \cdot \left(angle \cdot {a}^{2}\right)\right)} \]

   if 2.00000000000000004e-87 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 44.4%

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

      1. associate-*l* 44.4%

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

      2. associate-*l* 44.4%

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

   3. Simplified 44.4%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

      3. difference-of-squares 51.8%

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

   6. Applied egg-rr 51.8%

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

   7. Taylor expanded in angle around 0 47.1%

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

      1. associate-*r* 47.1%

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

   9. Simplified 47.1%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-87}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) - \pi \cdot \left(angle \cdot {a}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 52.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (+
   (* 0.011111111111111112 (* angle_m (* PI (pow b 2.0))))
   (*
    a
    (+
     (* -0.011111111111111112 (* a (* angle_m PI)))
     (* 0.011111111111111112 (* angle_m (* PI (- b b)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((0.011111111111111112 * (angle_m * (((double) M_PI) * pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * ((double) M_PI)))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - b)))))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((0.011111111111111112 * (angle_m * (Math.PI * Math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * Math.PI))) + (0.011111111111111112 * (angle_m * (Math.PI * (b - b)))))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((0.011111111111111112 * (angle_m * (math.pi * math.pow(b, 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * math.pi))) + (0.011111111111111112 * (angle_m * (math.pi * (b - b)))))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * (b ^ 2.0)))) + Float64(a * Float64(Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - b))))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((0.011111111111111112 * (angle_m * (pi * (b ^ 2.0)))) + (a * ((-0.011111111111111112 * (a * (angle_m * pi))) + (0.011111111111111112 * (angle_m * (pi * (b - b)))))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in angle around 0 51.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

7. Applied egg-rr 55.0%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

8. Taylor expanded in a around 0 56.6%

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

9. Final simplification 56.6%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot {b}^{2}\right)\right) + a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - b\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 23: 54.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   2.0
   (*
    (cos (* (/ angle_m 180.0) PI))
    (* (* (+ a b) (- b a)) (* angle_m (* 0.005555555555555556 PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (((a + b) * (b - a)) * (angle_m * (0.005555555555555556 * ((double) M_PI))))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (((a + b) * (b - a)) * (angle_m * (0.005555555555555556 * Math.PI)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (((a + b) * (b - a)) * (angle_m * (0.005555555555555556 * math.pi)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(angle_m * Float64(0.005555555555555556 * pi))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (2.0 * (cos(((angle_m / 180.0) * pi)) * (((a + b) * (b - a)) * (angle_m * (0.005555555555555556 * pi)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. associate-*l* 54.5%

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

3. Simplified 54.5%

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

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

6. Applied egg-rr 58.0%

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

7. Taylor expanded in angle around 0 56.4%

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

   1. *-commutative 56.4%

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

   2. associate-*r* 56.5%

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

9. Simplified 56.5%

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

10. Final simplification 56.5%

    \[\leadsto 2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 24: 52.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) - \pi \cdot \left(angle\_m \cdot {a}^{2}\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   0.011111111111111112
   (- (* b (* angle_m (* PI b))) (* PI (* angle_m (pow a 2.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((b * (angle_m * (((double) M_PI) * b))) - (((double) M_PI) * (angle_m * pow(a, 2.0)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((b * (angle_m * (Math.PI * b))) - (Math.PI * (angle_m * Math.pow(a, 2.0)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * ((b * (angle_m * (math.pi * b))) - (math.pi * (angle_m * math.pow(a, 2.0)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(b * Float64(angle_m * Float64(pi * b))) - Float64(pi * Float64(angle_m * (a ^ 2.0))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((b * (angle_m * (pi * b))) - (pi * (angle_m * (a ^ 2.0)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(b * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Pi * N[(angle$95$m * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in angle around 0 51.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

7. Applied egg-rr 55.0%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

8. Taylor expanded in b around 0 55.7%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(angle \cdot \left(b \cdot \pi\right) + angle \cdot \left(\pi \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 55.7%

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

   2. mul-1-neg 55.7%

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

   3. unsub-neg 55.7%

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

10. Simplified 55.7%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot \left(\pi \cdot \left(0 + b\right)\right)\right) - \pi \cdot \left(angle \cdot {a}^{2}\right)\right)} \]

11. Final simplification 55.7%

    \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) - \pi \cdot \left(angle \cdot {a}^{2}\right)\right) \]
12. Add Preprocessing

Alternative 25: 54.1% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* PI (* (+ a b) (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((a + b) * (b - a)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((a + b) * (b - a)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((a + b) * (b - a)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a + b) * Float64(b - a))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((a + b) * (b - a)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in angle around 0 51.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

7. Applied egg-rr 55.0%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

8. Final simplification 55.0%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 26: 21.5% accurate, 38.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a + b\right) \cdot \left(a \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* (+ a b) (* a PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (a * ((double) M_PI)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (a * Math.PI))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (a * math.pi))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a + b) * Float64(a * pi)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (a * pi))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a + b), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in angle around 0 51.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

7. Applied egg-rr 55.0%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]
8. Step-by-step derivation

   1. associate-*r* 55.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \]

   2. sub-neg 55.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

   3. distribute-lft-in 51.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \left(-a\right)\right)}\right) \]

   4. add-sqr-sqrt 24.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}\right)\right) \]

   5. sqrt-unprod 42.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right) \]

   6. sqr-neg 42.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \sqrt{\color{blue}{a \cdot a}}\right)\right) \]

   7. sqrt-unprod 18.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}\right)\right) \]

   8. add-sqr-sqrt 34.1%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{a}\right)\right) \]

9. Applied egg-rr 34.1%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot a\right)}\right) \]
10. Step-by-step derivation

    1. distribute-lft-out 36.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

11. Simplified 36.9%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

12. Taylor expanded in b around 0 22.5%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(a \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]

13. Final simplification 22.5%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot \left(a \cdot \pi\right)\right)\right) \]
14. Add Preprocessing

Alternative 27: 37.2% accurate, 38.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a + b\right) \cdot \left(\pi \cdot b\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* (+ a b) (* PI b))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (((double) M_PI) * b))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (Math.PI * b))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (math.pi * b))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(a + b) * Float64(pi * b)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * ((a + b) * (pi * b))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(a + b), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. *-commutative 54.5%

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

   3. associate-*l* 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in angle around 0 51.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 54.5%

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

   2. unpow2 54.5%

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

   3. difference-of-squares 58.0%

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

7. Applied egg-rr 55.0%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]
8. Step-by-step derivation

   1. associate-*r* 55.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \]

   2. sub-neg 55.0%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b + \left(-a\right)\right)}\right)\right) \]

   3. distribute-lft-in 51.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \left(-a\right)\right)}\right) \]

   4. add-sqr-sqrt 24.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}\right)\right) \]

   5. sqrt-unprod 42.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}\right)\right) \]

   6. sqr-neg 42.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \sqrt{\color{blue}{a \cdot a}}\right)\right) \]

   7. sqrt-unprod 18.2%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}\right)\right) \]

   8. add-sqr-sqrt 34.1%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot \color{blue}{a}\right)\right) \]

9. Applied egg-rr 34.1%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot b + \left(\pi \cdot \left(b + a\right)\right) \cdot a\right)}\right) \]
10. Step-by-step derivation

    1. distribute-lft-out 36.9%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

11. Simplified 36.9%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b + a\right)\right)}\right) \]

12. Taylor expanded in b around inf 36.7%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(b \cdot \pi\right)} \cdot \left(b + a\right)\right)\right) \]
13. Step-by-step derivation

    1. *-commutative 36.7%

       \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot b\right)} \cdot \left(b + a\right)\right)\right) \]

14. Simplified 36.7%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\left(\pi \cdot b\right)} \cdot \left(b + a\right)\right)\right) \]

15. Final simplification 36.7%

    \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\left(a + b\right) \cdot \left(\pi \cdot b\right)\right)\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))