ab-angle->ABCF B

Percentage Accurate: 54.2% → 67.4%

Time: 30.0s

Alternatives: 13

Speedup: 32.2×

• 33.3% 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: 54.2% 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: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ t_1 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(b, 2 \cdot \left(b \cdot t\_0\right), -2 \cdot \left(a \cdot \left(t\_0 \cdot a\right)\right)\right) \cdot t\_1\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+153}:\\ \;\;\;\;t\_1 \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle_m PI))))
        (t_1 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+51)
      (* (fma b (* 2.0 (* b t_0)) (* -2.0 (* a (* t_0 a)))) t_1)
      (if (<= (/ angle_m 180.0) 1e+153)
        (* t_1 (* (* (+ b a) (- a b)) (* 2.0 t_0)))
        (*
         (* t_0 (* 2.0 (* (+ b a) (- b a))))
         (pow (cbrt (cos (* PI (* angle_m 0.005555555555555556)))) 3.0)))))))

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((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double t_1 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle_m / 180.0) <= 2e+51) {
		tmp = fma(b, (2.0 * (b * t_0)), (-2.0 * (a * (t_0 * a)))) * t_1;
	} else if ((angle_m / 180.0) <= 1e+153) {
		tmp = t_1 * (((b + a) * (a - b)) * (2.0 * t_0));
	} else {
		tmp = (t_0 * (2.0 * ((b + a) * (b - a)))) * pow(cbrt(cos((((double) M_PI) * (angle_m * 0.005555555555555556)))), 3.0);
	}
	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(0.005555555555555556 * Float64(angle_m * pi)))
	t_1 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+51)
		tmp = Float64(fma(b, Float64(2.0 * Float64(b * t_0)), Float64(-2.0 * Float64(a * Float64(t_0 * a)))) * t_1);
	elseif (Float64(angle_m / 180.0) <= 1e+153)
		tmp = Float64(t_1 * Float64(Float64(Float64(b + a) * Float64(a - b)) * Float64(2.0 * t_0)));
	else
		tmp = Float64(Float64(t_0 * Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))) * (cbrt(cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 3.0));
	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[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+51], N[(N[(b * N[(2.0 * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a * N[(t$95$0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+153], N[(t$95$1 * N[(N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e51

   1. Initial program 55.1%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. difference-of-squares 57.8%

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

   4. Applied egg-rr 57.8%

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

   5. Taylor expanded in b around 0 63.1%

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

      1. +-commutative 63.1%

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

      2. fma-define 64.6%

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

      3. distribute-lft-out 64.6%

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

      4. *-commutative 64.6%

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

      5. distribute-rgt1-in 64.6%

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

      6. metadata-eval 64.6%

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

      7. mul0-lft 64.6%

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

      8. distribute-lft-out 64.6%

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

      9. +-rgt-identity 64.6%

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

   7. Simplified 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*r* 63.3%

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

      3. *-commutative 63.3%

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

      4. metadata-eval 63.3%

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

      5. div-inv 62.6%

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

      6. *-commutative 62.6%

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

      7. unpow2 62.6%

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

      8. associate-*r* 70.7%

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

      9. *-commutative 70.7%

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

      10. div-inv 71.3%

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

      11. metadata-eval 71.3%

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

      12. *-commutative 71.3%

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

      13. associate-*r* 72.6%

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

   9. Applied egg-rr 72.6%

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

   if 2e51 < (/.f64 angle #s(literal 180 binary64)) < 1e153

   1. Initial program 43.4%

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

   3. Taylor expanded in b around 0 23.5%

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

   5. Simplified 49.9%

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

      1. unpow2 49.9%

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

      2. unpow2 49.9%

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

      3. difference-of-squares 49.9%

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

   7. Applied egg-rr 49.9%

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

   if 1e153 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 39.5%

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

      1. unpow2 39.5%

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

      2. unpow2 39.5%

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

      3. difference-of-squares 42.3%

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

   4. Applied egg-rr 42.3%

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

      1. div-inv 40.1%

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

      2. metadata-eval 40.1%

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

      3. add-cube-cbrt 40.1%

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

      4. pow3 40.1%

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

   6. Applied egg-rr 40.1%

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

   7. Taylor expanded in angle around inf 47.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(b, 2 \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right), -2 \cdot \left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+153}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ t_1 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(b, 2 \cdot \left(b \cdot t\_1\right), -2 \cdot \left(a \cdot \left(t\_1 \cdot a\right)\right)\right) \cdot \cos t\_0\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+153}:\\ \;\;\;\;\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle_m PI))) (t_1 (sin t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+64)
      (* (fma b (* 2.0 (* b t_1)) (* -2.0 (* a (* t_1 a)))) (cos t_0))
      (if (<= (/ angle_m 180.0) 1e+153)
        (* (cos (* (/ angle_m 180.0) PI)) (* (* (+ b a) (- a b)) (* 2.0 t_1)))
        (*
         (* t_1 (* 2.0 (* (+ b a) (- b a))))
         (pow (cbrt (cos (* PI (* angle_m 0.005555555555555556)))) 3.0)))))))

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 = 0.005555555555555556 * (angle_m * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 2e+64) {
		tmp = fma(b, (2.0 * (b * t_1)), (-2.0 * (a * (t_1 * a)))) * cos(t_0);
	} else if ((angle_m / 180.0) <= 1e+153) {
		tmp = cos(((angle_m / 180.0) * ((double) M_PI))) * (((b + a) * (a - b)) * (2.0 * t_1));
	} else {
		tmp = (t_1 * (2.0 * ((b + a) * (b - a)))) * pow(cbrt(cos((((double) M_PI) * (angle_m * 0.005555555555555556)))), 3.0);
	}
	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 = Float64(0.005555555555555556 * Float64(angle_m * pi))
	t_1 = sin(t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+64)
		tmp = Float64(fma(b, Float64(2.0 * Float64(b * t_1)), Float64(-2.0 * Float64(a * Float64(t_1 * a)))) * cos(t_0));
	elseif (Float64(angle_m / 180.0) <= 1e+153)
		tmp = Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(Float64(b + a) * Float64(a - b)) * Float64(2.0 * t_1)));
	else
		tmp = Float64(Float64(t_1 * Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))) * (cbrt(cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 3.0));
	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[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+64], N[(N[(b * N[(2.0 * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a * N[(t$95$1 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+153], N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000004e64

   1. Initial program 55.0%

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

      1. unpow2 55.0%

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

      2. unpow2 55.0%

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

      3. difference-of-squares 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in b around 0 62.7%

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

      1. +-commutative 62.7%

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

      2. fma-define 64.2%

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

      3. distribute-lft-out 64.2%

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

      4. *-commutative 64.2%

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

      5. distribute-rgt1-in 64.2%

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

      6. metadata-eval 64.2%

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

      7. mul0-lft 64.2%

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

      8. distribute-lft-out 64.2%

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

      9. +-rgt-identity 64.2%

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

   7. Simplified 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-*r* 63.0%

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

      3. *-commutative 63.0%

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

      4. metadata-eval 63.0%

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

      5. div-inv 62.4%

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

      6. *-commutative 62.4%

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

      7. unpow2 62.4%

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

      8. associate-*r* 70.3%

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

      9. *-commutative 70.3%

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

      10. div-inv 71.0%

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

      11. metadata-eval 71.0%

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

      12. *-commutative 71.0%

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

      13. associate-*r* 72.2%

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

   9. Applied egg-rr 72.2%

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

   10. Taylor expanded in angle around inf 71.2%

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

   if 2.00000000000000004e64 < (/.f64 angle #s(literal 180 binary64)) < 1e153

   1. Initial program 42.9%

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

   3. Taylor expanded in b around 0 20.4%

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

   5. Simplified 51.7%

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

      1. unpow2 51.7%

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

      2. unpow2 51.7%

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

      3. difference-of-squares 51.7%

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

   7. Applied egg-rr 51.7%

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

   if 1e153 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 39.5%

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

      1. unpow2 39.5%

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

      2. unpow2 39.5%

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

      3. difference-of-squares 42.3%

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

   4. Applied egg-rr 42.3%

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

      1. div-inv 40.1%

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

      2. metadata-eval 40.1%

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

      3. add-cube-cbrt 40.1%

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

      4. pow3 40.1%

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

   6. Applied egg-rr 40.1%

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

   7. Taylor expanded in angle around inf 47.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(b, 2 \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right), -2 \cdot \left(a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+153}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle\_m \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+153}:\\ \;\;\;\;\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle_m PI)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+51)
      (* (- a b) (* (* angle_m -0.011111111111111112) (* PI (+ b a))))
      (if (<= (/ angle_m 180.0) 1e+153)
        (* (cos (* (/ angle_m 180.0) PI)) (* (* (+ b a) (- a b)) (* 2.0 t_0)))
        (*
         (* t_0 (* 2.0 (* (+ b a) (- b a))))
         (pow (cbrt (cos (* PI (* angle_m 0.005555555555555556)))) 3.0)))))))

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((0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 2e+51) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (((double) M_PI) * (b + a)));
	} else if ((angle_m / 180.0) <= 1e+153) {
		tmp = cos(((angle_m / 180.0) * ((double) M_PI))) * (((b + a) * (a - b)) * (2.0 * t_0));
	} else {
		tmp = (t_0 * (2.0 * ((b + a) * (b - a)))) * pow(cbrt(cos((((double) M_PI) * (angle_m * 0.005555555555555556)))), 3.0);
	}
	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 t_0 = Math.sin((0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if ((angle_m / 180.0) <= 2e+51) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (Math.PI * (b + a)));
	} else if ((angle_m / 180.0) <= 1e+153) {
		tmp = Math.cos(((angle_m / 180.0) * Math.PI)) * (((b + a) * (a - b)) * (2.0 * t_0));
	} else {
		tmp = (t_0 * (2.0 * ((b + a) * (b - a)))) * Math.pow(Math.cbrt(Math.cos((Math.PI * (angle_m * 0.005555555555555556)))), 3.0);
	}
	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(0.005555555555555556 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+51)
		tmp = Float64(Float64(a - b) * Float64(Float64(angle_m * -0.011111111111111112) * Float64(pi * Float64(b + a))));
	elseif (Float64(angle_m / 180.0) <= 1e+153)
		tmp = Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(Float64(Float64(b + a) * Float64(a - b)) * Float64(2.0 * t_0)));
	else
		tmp = Float64(Float64(t_0 * Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))) * (cbrt(cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 3.0));
	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[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+51], N[(N[(a - b), $MachinePrecision] * N[(N[(angle$95$m * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+153], N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e51

   1. Initial program 55.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in angle around 0 54.0%

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

      1. unpow2 16.0%

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

      2. unpow2 16.0%

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

      3. difference-of-squares 17.0%

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

   7. Applied egg-rr 56.2%

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

      1. associate-*r* 56.2%

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

      2. associate-*r* 56.2%

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

      3. associate-*r* 68.7%

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

      4. *-commutative 68.7%

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

   9. Applied egg-rr 68.7%

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

   if 2e51 < (/.f64 angle #s(literal 180 binary64)) < 1e153

   1. Initial program 43.4%

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

   3. Taylor expanded in b around 0 23.5%

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

   5. Simplified 49.9%

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

      1. unpow2 49.9%

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

      2. unpow2 49.9%

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

      3. difference-of-squares 49.9%

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

   7. Applied egg-rr 49.9%

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

   if 1e153 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 39.5%

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

      1. unpow2 39.5%

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

      2. unpow2 39.5%

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

      3. difference-of-squares 42.3%

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

   4. Applied egg-rr 42.3%

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

      1. div-inv 40.1%

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

      2. metadata-eval 40.1%

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

      3. add-cube-cbrt 40.1%

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

      4. pow3 40.1%

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

   6. Applied egg-rr 40.1%

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

   7. Taylor expanded in angle around inf 47.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+153}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ t_1 := \left(b + a\right) \cdot \left(a - b\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle\_m \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+153}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_0 \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (cos (* (/ angle_m 180.0) PI))) (t_1 (* (+ b a) (- a b))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+51)
      (* (- a b) (* (* angle_m -0.011111111111111112) (* PI (+ b a))))
      (if (<= (/ angle_m 180.0) 1e+153)
        (* t_0 (* t_1 (* 2.0 (sin (* 0.005555555555555556 (* angle_m PI))))))
        (*
         t_1
         (* 2.0 (* t_0 (sin (* (* angle_m PI) -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 t_0 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double t_1 = (b + a) * (a - b);
	double tmp;
	if ((angle_m / 180.0) <= 2e+51) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (((double) M_PI) * (b + a)));
	} else if ((angle_m / 180.0) <= 1e+153) {
		tmp = t_0 * (t_1 * (2.0 * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = t_1 * (2.0 * (t_0 * sin(((angle_m * ((double) M_PI)) * -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 t_0 = Math.cos(((angle_m / 180.0) * Math.PI));
	double t_1 = (b + a) * (a - b);
	double tmp;
	if ((angle_m / 180.0) <= 2e+51) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (Math.PI * (b + a)));
	} else if ((angle_m / 180.0) <= 1e+153) {
		tmp = t_0 * (t_1 * (2.0 * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))));
	} else {
		tmp = t_1 * (2.0 * (t_0 * Math.sin(((angle_m * Math.PI) * -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):
	t_0 = math.cos(((angle_m / 180.0) * math.pi))
	t_1 = (b + a) * (a - b)
	tmp = 0
	if (angle_m / 180.0) <= 2e+51:
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (math.pi * (b + a)))
	elif (angle_m / 180.0) <= 1e+153:
		tmp = t_0 * (t_1 * (2.0 * math.sin((0.005555555555555556 * (angle_m * math.pi)))))
	else:
		tmp = t_1 * (2.0 * (t_0 * math.sin(((angle_m * math.pi) * -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)
	t_0 = cos(Float64(Float64(angle_m / 180.0) * pi))
	t_1 = Float64(Float64(b + a) * Float64(a - b))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+51)
		tmp = Float64(Float64(a - b) * Float64(Float64(angle_m * -0.011111111111111112) * Float64(pi * Float64(b + a))));
	elseif (Float64(angle_m / 180.0) <= 1e+153)
		tmp = Float64(t_0 * Float64(t_1 * Float64(2.0 * sin(Float64(0.005555555555555556 * Float64(angle_m * pi))))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * sin(Float64(Float64(angle_m * pi) * -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)
	t_0 = cos(((angle_m / 180.0) * pi));
	t_1 = (b + a) * (a - b);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+51)
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (pi * (b + a)));
	elseif ((angle_m / 180.0) <= 1e+153)
		tmp = t_0 * (t_1 * (2.0 * sin((0.005555555555555556 * (angle_m * pi)))));
	else
		tmp = t_1 * (2.0 * (t_0 * sin(((angle_m * pi) * -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_] := Block[{t$95$0 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+51], N[(N[(a - b), $MachinePrecision] * N[(N[(angle$95$m * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+153], N[(t$95$0 * N[(t$95$1 * N[(2.0 * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e51

   1. Initial program 55.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in angle around 0 54.0%

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

      1. unpow2 16.0%

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

      2. unpow2 16.0%

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

      3. difference-of-squares 17.0%

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

   7. Applied egg-rr 56.2%

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

      1. associate-*r* 56.2%

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

      2. associate-*r* 56.2%

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

      3. associate-*r* 68.7%

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

      4. *-commutative 68.7%

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

   9. Applied egg-rr 68.7%

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

   if 2e51 < (/.f64 angle #s(literal 180 binary64)) < 1e153

   1. Initial program 43.4%

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

   3. Taylor expanded in b around 0 23.5%

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

   5. Simplified 49.9%

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

      1. unpow2 49.9%

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

      2. unpow2 49.9%

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

      3. difference-of-squares 49.9%

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

   7. Applied egg-rr 49.9%

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

   if 1e153 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 39.5%

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

   3. Simplified 44.1%

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

      1. unpow2 23.9%

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

      2. unpow2 23.9%

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

      3. difference-of-squares 29.6%

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

   6. Applied egg-rr 49.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+153}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ t_1 := 2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-134}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle\_m \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\cos t\_0 \cdot \left(t\_1 \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sin t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (/ angle_m 180.0) PI)) (t_1 (* 2.0 (* (+ b a) (- b a)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-134)
      (* (- a b) (* (* angle_m -0.011111111111111112) (* PI (+ b a))))
      (if (<= (/ angle_m 180.0) 4e+174)
        (* (cos t_0) (* t_1 (sin (* PI (* angle_m 0.005555555555555556)))))
        (* t_1 (sin 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 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle_m / 180.0) <= 1e-134) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (((double) M_PI) * (b + a)));
	} else if ((angle_m / 180.0) <= 4e+174) {
		tmp = cos(t_0) * (t_1 * sin((((double) M_PI) * (angle_m * 0.005555555555555556))));
	} else {
		tmp = t_1 * sin(t_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 t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = 2.0 * ((b + a) * (b - a));
	double tmp;
	if ((angle_m / 180.0) <= 1e-134) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (Math.PI * (b + a)));
	} else if ((angle_m / 180.0) <= 4e+174) {
		tmp = Math.cos(t_0) * (t_1 * Math.sin((Math.PI * (angle_m * 0.005555555555555556))));
	} else {
		tmp = t_1 * Math.sin(t_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):
	t_0 = (angle_m / 180.0) * math.pi
	t_1 = 2.0 * ((b + a) * (b - a))
	tmp = 0
	if (angle_m / 180.0) <= 1e-134:
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (math.pi * (b + a)))
	elif (angle_m / 180.0) <= 4e+174:
		tmp = math.cos(t_0) * (t_1 * math.sin((math.pi * (angle_m * 0.005555555555555556))))
	else:
		tmp = t_1 * math.sin(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 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-134)
		tmp = Float64(Float64(a - b) * Float64(Float64(angle_m * -0.011111111111111112) * Float64(pi * Float64(b + a))));
	elseif (Float64(angle_m / 180.0) <= 4e+174)
		tmp = Float64(cos(t_0) * Float64(t_1 * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))));
	else
		tmp = Float64(t_1 * sin(t_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)
	t_0 = (angle_m / 180.0) * pi;
	t_1 = 2.0 * ((b + a) * (b - a));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e-134)
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (pi * (b + a)));
	elseif ((angle_m / 180.0) <= 4e+174)
		tmp = cos(t_0) * (t_1 * sin((pi * (angle_m * 0.005555555555555556))));
	else
		tmp = t_1 * sin(t_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_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-134], N[(N[(a - b), $MachinePrecision] * N[(N[(angle$95$m * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+174], N[(N[Cos[t$95$0], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000004e-134

   1. Initial program 47.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in angle around 0 48.0%

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

      1. unpow2 17.3%

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

      2. unpow2 17.3%

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

      3. difference-of-squares 18.6%

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

   7. Applied egg-rr 50.1%

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

      1. associate-*r* 50.0%

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

      2. associate-*r* 50.1%

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

      3. associate-*r* 65.8%

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

      4. *-commutative 65.8%

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

   9. Applied egg-rr 65.8%

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

   if 1.00000000000000004e-134 < (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000028e174

   1. Initial program 69.8%

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

      1. unpow2 69.8%

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

      2. unpow2 69.8%

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

      3. difference-of-squares 71.3%

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

   4. Applied egg-rr 71.3%

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

      1. *-commutative 71.3%

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

      2. div-inv 72.8%

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

      3. metadata-eval 72.8%

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

      4. *-commutative 72.8%

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

      5. associate-*r* 67.4%

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

      6. *-commutative 67.4%

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

      7. *-commutative 67.4%

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

      8. associate-*r* 72.8%

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

   6. Applied egg-rr 72.8%

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

   if 4.00000000000000028e174 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 35.1%

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

      1. unpow2 35.1%

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

      2. unpow2 35.1%

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

      3. difference-of-squares 38.4%

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

   4. Applied egg-rr 38.4%

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

   5. Taylor expanded in angle around 0 51.6%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-134}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+174}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+76}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle\_m \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\\ \end{array} \end{array} \]

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+76)
    (* (- a b) (* (* angle_m -0.011111111111111112) (* PI (+ b a))))
    (* (* 2.0 (* (+ b a) (- b a))) (sin (* (/ angle_m 180.0) 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) <= 1e+76) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (((double) M_PI) * (b + a)));
	} else {
		tmp = (2.0 * ((b + a) * (b - a))) * sin(((angle_m / 180.0) * ((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) <= 1e+76) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (Math.PI * (b + a)));
	} else {
		tmp = (2.0 * ((b + a) * (b - a))) * Math.sin(((angle_m / 180.0) * 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) <= 1e+76:
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (math.pi * (b + a)))
	else:
		tmp = (2.0 * ((b + a) * (b - a))) * math.sin(((angle_m / 180.0) * 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) <= 1e+76)
		tmp = Float64(Float64(a - b) * Float64(Float64(angle_m * -0.011111111111111112) * Float64(pi * Float64(b + a))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))) * sin(Float64(Float64(angle_m / 180.0) * 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) <= 1e+76)
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * (pi * (b + a)));
	else
		tmp = (2.0 * ((b + a) * (b - a))) * sin(((angle_m / 180.0) * 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], 1e+76], N[(N[(a - b), $MachinePrecision] * N[(N[(angle$95$m * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e76

   1. Initial program 54.7%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in angle around 0 54.0%

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

      1. unpow2 16.8%

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

      2. unpow2 16.8%

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

      3. difference-of-squares 17.8%

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

   7. Applied egg-rr 56.1%

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

      1. associate-*r* 56.1%

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

      2. associate-*r* 56.2%

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

      3. associate-*r* 67.9%

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

      4. *-commutative 67.9%

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

   9. Applied egg-rr 67.9%

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

   if 1e76 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 41.4%

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

      1. unpow2 41.4%

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

      2. unpow2 41.4%

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

      3. difference-of-squares 43.3%

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

   4. Applied egg-rr 43.3%

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

   5. Taylor expanded in angle around 0 50.1%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+76}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(b + a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.5 \cdot 10^{-114}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle\_m \cdot -0.011111111111111112\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a - b\right) \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* PI (+ b a))))
   (*
    angle_s
    (if (<= angle_m 1.5e-114)
      (* (- a b) (* (* angle_m -0.011111111111111112) t_0))
      (* -0.011111111111111112 (* angle_m (* (- a b) 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 = ((double) M_PI) * (b + a);
	double tmp;
	if (angle_m <= 1.5e-114) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * t_0);
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * t_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 t_0 = Math.PI * (b + a);
	double tmp;
	if (angle_m <= 1.5e-114) {
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * t_0);
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * t_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):
	t_0 = math.pi * (b + a)
	tmp = 0
	if angle_m <= 1.5e-114:
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * t_0)
	else:
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * 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 = Float64(pi * Float64(b + a))
	tmp = 0.0
	if (angle_m <= 1.5e-114)
		tmp = Float64(Float64(a - b) * Float64(Float64(angle_m * -0.011111111111111112) * t_0));
	else
		tmp = Float64(-0.011111111111111112 * Float64(angle_m * Float64(Float64(a - b) * t_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)
	t_0 = pi * (b + a);
	tmp = 0.0;
	if (angle_m <= 1.5e-114)
		tmp = (a - b) * ((angle_m * -0.011111111111111112) * t_0);
	else
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * t_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_] := Block[{t$95$0 = N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 1.5e-114], N[(N[(a - b), $MachinePrecision] * N[(N[(angle$95$m * -0.011111111111111112), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(angle$95$m * N[(N[(a - b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.50000000000000008e-114

   1. Initial program 48.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in angle around 0 48.3%

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

      1. unpow2 17.2%

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

      2. unpow2 17.2%

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

      3. difference-of-squares 18.5%

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

   7. Applied egg-rr 50.4%

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

      1. associate-*r* 50.4%

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

      2. associate-*r* 50.4%

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

      3. associate-*r* 66.1%

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

      4. *-commutative 66.1%

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

   9. Applied egg-rr 66.1%

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

   if 1.50000000000000008e-114 < angle

   1. Initial program 58.6%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in angle around 0 50.2%

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

      1. unpow2 24.0%

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

      2. unpow2 24.0%

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

      3. difference-of-squares 26.1%

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

   7. Applied egg-rr 53.3%

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

      1. associate-*r* 53.4%

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

      2. *-commutative 53.4%

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

   9. Applied egg-rr 53.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.5 \cdot 10^{-114}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 7.5e-20)
    (* -0.011111111111111112 (* (- a b) (* (* angle_m PI) (+ b a))))
    (* -0.011111111111111112 (* angle_m (* (- a b) (* PI (+ 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 <= 7.5e-20) {
		tmp = -0.011111111111111112 * ((a - b) * ((angle_m * ((double) M_PI)) * (b + a)));
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (((double) M_PI) * (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 <= 7.5e-20) {
		tmp = -0.011111111111111112 * ((a - b) * ((angle_m * Math.PI) * (b + a)));
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (Math.PI * (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 <= 7.5e-20:
		tmp = -0.011111111111111112 * ((a - b) * ((angle_m * math.pi) * (b + a)))
	else:
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (math.pi * (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 (angle_m <= 7.5e-20)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(Float64(angle_m * pi) * Float64(b + a))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(angle_m * Float64(Float64(a - b) * Float64(pi * 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 <= 7.5e-20)
		tmp = -0.011111111111111112 * ((a - b) * ((angle_m * pi) * (b + a)));
	else
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (pi * (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[angle$95$m, 7.5e-20], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(angle$95$m * N[(N[(a - b), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 7.49999999999999981e-20

   1. Initial program 54.3%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in angle around 0 54.4%

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

      1. unpow2 16.7%

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

      2. unpow2 16.7%

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

      3. difference-of-squares 17.8%

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

   7. Applied egg-rr 56.8%

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

      1. associate-*r* 56.8%

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

      2. associate-*r* 70.0%

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

   9. Applied egg-rr 70.0%

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

   if 7.49999999999999981e-20 < angle

   1. Initial program 46.4%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in angle around 0 35.2%

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

      1. unpow2 27.6%

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

      2. unpow2 27.6%

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

      3. difference-of-squares 30.3%

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

   7. Applied egg-rr 38.0%

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

      1. associate-*r* 38.0%

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

      2. *-commutative 38.0%

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

   9. Applied egg-rr 38.0%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(b + a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.6 \cdot 10^{-131}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle\_m \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a - b\right) \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* PI (+ b a))))
   (*
    angle_s
    (if (<= angle_m 2.6e-131)
      (* -0.011111111111111112 (* (- a b) (* angle_m t_0)))
      (* -0.011111111111111112 (* angle_m (* (- a b) 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 = ((double) M_PI) * (b + a);
	double tmp;
	if (angle_m <= 2.6e-131) {
		tmp = -0.011111111111111112 * ((a - b) * (angle_m * t_0));
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * t_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 t_0 = Math.PI * (b + a);
	double tmp;
	if (angle_m <= 2.6e-131) {
		tmp = -0.011111111111111112 * ((a - b) * (angle_m * t_0));
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * t_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):
	t_0 = math.pi * (b + a)
	tmp = 0
	if angle_m <= 2.6e-131:
		tmp = -0.011111111111111112 * ((a - b) * (angle_m * t_0))
	else:
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * 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 = Float64(pi * Float64(b + a))
	tmp = 0.0
	if (angle_m <= 2.6e-131)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle_m * t_0)));
	else
		tmp = Float64(-0.011111111111111112 * Float64(angle_m * Float64(Float64(a - b) * t_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)
	t_0 = pi * (b + a);
	tmp = 0.0;
	if (angle_m <= 2.6e-131)
		tmp = -0.011111111111111112 * ((a - b) * (angle_m * t_0));
	else
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * t_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_] := Block[{t$95$0 = N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.6e-131], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(angle$95$m * N[(N[(a - b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.59999999999999996e-131

   1. Initial program 47.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in angle around 0 48.0%

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

      1. unpow2 17.3%

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

      2. unpow2 17.3%

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

      3. difference-of-squares 18.6%

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

   7. Applied egg-rr 50.1%

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

      1. associate-*r* 50.1%

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

      2. associate-*r* 65.4%

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

   9. Applied egg-rr 65.4%

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

   if 2.59999999999999996e-131 < angle

   1. Initial program 59.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in angle around 0 50.7%

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

      1. unpow2 23.8%

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

      2. unpow2 23.8%

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

      3. difference-of-squares 25.8%

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

   7. Applied egg-rr 53.8%

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

      1. associate-*r* 53.8%

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

      2. *-commutative 53.8%

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

   9. Applied egg-rr 53.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.6 \cdot 10^{-131}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.2% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(a - b\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= b 6.2e-78)
    (* -0.011111111111111112 (* angle_m (* (- a b) (* PI a))))
    (* -0.011111111111111112 (* angle_m (* (- a b) (* b 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 (b <= 6.2e-78) {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (((double) M_PI) * a)));
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (b * ((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 (b <= 6.2e-78) {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (Math.PI * a)));
	} else {
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (b * 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 b <= 6.2e-78:
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (math.pi * a)))
	else:
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (b * 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 (b <= 6.2e-78)
		tmp = Float64(-0.011111111111111112 * Float64(angle_m * Float64(Float64(a - b) * Float64(pi * a))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(angle_m * Float64(Float64(a - b) * Float64(b * 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 <= 6.2e-78)
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (pi * a)));
	else
		tmp = -0.011111111111111112 * (angle_m * ((a - b) * (b * 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[b, 6.2e-78], N[(-0.011111111111111112 * N[(angle$95$m * N[(N[(a - b), $MachinePrecision] * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(angle$95$m * N[(N[(a - b), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000035e-78

   1. Initial program 52.8%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in angle around 0 53.0%

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

      1. unpow2 21.5%

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

      2. unpow2 21.5%

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

      3. difference-of-squares 23.2%

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

   7. Applied egg-rr 55.8%

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

      1. associate-*r* 55.9%

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

      2. *-commutative 55.9%

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

   9. Applied egg-rr 55.9%

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

   10. Taylor expanded in a around inf 40.2%

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

   if 6.20000000000000035e-78 < b

   1. Initial program 50.4%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in angle around 0 39.5%

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

      1. unpow2 15.6%

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

      2. unpow2 15.6%

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

      3. difference-of-squares 16.9%

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

   7. Applied egg-rr 41.1%

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

      1. associate-*r* 41.2%

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

      2. *-commutative 41.2%

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

   9. Applied egg-rr 41.2%

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

   10. Taylor expanded in a around 0 36.1%

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

       1. *-commutative 36.1%

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

   12. Simplified 36.1%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 32.2× speedup

Math

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

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 (* (- a b) (* PI (+ 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 * ((a - b) * (((double) M_PI) * (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 * ((a - b) * (Math.PI * (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 * ((a - b) * (math.pi * (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(Float64(a - b) * Float64(pi * 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 * ((a - b) * (pi * (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[(N[(a - b), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

3. Simplified 51.9%

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

5. Taylor expanded in angle around 0 49.0%

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

   1. unpow2 19.8%

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

   2. unpow2 19.8%

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

   3. difference-of-squares 21.3%

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

7. Applied egg-rr 51.5%

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

   1. associate-*r* 51.5%

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

   2. *-commutative 51.5%

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

9. Applied egg-rr 51.5%

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

10. Final simplification 51.5%

    \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 12: 54.5% accurate, 32.2× speedup

Math

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

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 (* (+ b a) (- 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) {
	return angle_s * (-0.011111111111111112 * (angle_m * (((double) M_PI) * ((b + a) * (a - 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 * ((b + a) * (a - 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 * ((b + a) * (a - 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(pi * Float64(Float64(b + a) * Float64(a - 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 + a) * (a - 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[(Pi * N[(N[(b + a), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

3. Simplified 51.9%

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

5. Taylor expanded in angle around 0 49.0%

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

   1. unpow2 19.8%

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

   2. unpow2 19.8%

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

   3. difference-of-squares 21.3%

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

7. Applied egg-rr 51.5%

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

8. Final simplification 51.5%

   \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 13: 37.0% accurate, 38.1× speedup

Math

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

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 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 * ((a - b) * (((double) M_PI) * 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 * ((a - b) * (Math.PI * 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 * ((a - b) * (math.pi * 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(Float64(a - b) * Float64(pi * 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 * ((a - b) * (pi * 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[(N[(a - b), $MachinePrecision] * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

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

3. Simplified 51.9%

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

5. Taylor expanded in angle around 0 49.0%

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

   1. unpow2 19.8%

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

   2. unpow2 19.8%

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

   3. difference-of-squares 21.3%

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

7. Applied egg-rr 51.5%

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

   1. associate-*r* 51.5%

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

   2. *-commutative 51.5%

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

9. Applied egg-rr 51.5%

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

10. Taylor expanded in a around inf 33.1%

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

11. Final simplification 33.1%

    \[\leadsto -0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot a\right)\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))