ab-angle->ABCF B

Percentage Accurate: 54.4% → 67.7%

Time: 17.7s

Alternatives: 17

Speedup: 23.3×

• Could not converge on a ground truth

  1. reg0 = (bf "2.748716162706708287334986569851983678311e-168")
  2. reg1 = (bf "1.957993054601019012891461412526523709679e-283")
  3. reg2 = (bf "3.174546886836536485809152045434675976001e-302")

• 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: 54.4% 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: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\\ t_1 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\left(\cos t\_0 \cdot 2\right) \cdot \left(\left(a - b\_m\right) \cdot \left(\left(a + b\_m\right) \cdot t\_1\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b\_m, \sin \left(angle\_m \cdot \left(-0.005555555555555556 \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right) \cdot \left(-b\_m\right), t\_1 \cdot {a}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle\_m \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot \left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI -0.005555555555555556))) (t_1 (sin t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+17)
      (* (* (cos t_0) 2.0) (* (- a b_m) (* (+ a b_m) t_1)))
      (if (<= (/ angle_m 180.0) 2e+246)
        (*
         2.0
         (fma
          b_m
          (*
           (sin (* angle_m (* -0.005555555555555556 (pow (sqrt PI) 2.0))))
           (- b_m))
          (* t_1 (pow a 2.0))))
        (*
         (cos (* angle_m (/ PI -180.0)))
         (*
          2.0
          (*
           (sin (* angle_m (log (+ 1.0 (expm1 (* PI -0.005555555555555556))))))
           (* (- a b_m) (+ a b_m))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * -0.005555555555555556);
	double t_1 = sin(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 2e+17) {
		tmp = (cos(t_0) * 2.0) * ((a - b_m) * ((a + b_m) * t_1));
	} else if ((angle_m / 180.0) <= 2e+246) {
		tmp = 2.0 * fma(b_m, (sin((angle_m * (-0.005555555555555556 * pow(sqrt(((double) M_PI)), 2.0)))) * -b_m), (t_1 * pow(a, 2.0)));
	} else {
		tmp = cos((angle_m * (((double) M_PI) / -180.0))) * (2.0 * (sin((angle_m * log((1.0 + expm1((((double) M_PI) * -0.005555555555555556)))))) * ((a - b_m) * (a + b_m))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(angle_m * Float64(pi * -0.005555555555555556))
	t_1 = sin(t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+17)
		tmp = Float64(Float64(cos(t_0) * 2.0) * Float64(Float64(a - b_m) * Float64(Float64(a + b_m) * t_1)));
	elseif (Float64(angle_m / 180.0) <= 2e+246)
		tmp = Float64(2.0 * fma(b_m, Float64(sin(Float64(angle_m * Float64(-0.005555555555555556 * (sqrt(pi) ^ 2.0)))) * Float64(-b_m)), Float64(t_1 * (a ^ 2.0))));
	else
		tmp = Float64(cos(Float64(angle_m * Float64(pi / -180.0))) * Float64(2.0 * Float64(sin(Float64(angle_m * log(Float64(1.0 + expm1(Float64(pi * -0.005555555555555556)))))) * Float64(Float64(a - b_m) * Float64(a + b_m)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * -0.005555555555555556), $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+17], N[(N[(N[Cos[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a - b$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+246], N[(2.0 * N[(b$95$m * N[(N[Sin[N[(angle$95$m * N[(-0.005555555555555556 * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-b$95$m)), $MachinePrecision] + N[(t$95$1 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[Sin[N[(angle$95$m * N[Log[N[(1.0 + N[(Exp[N[(Pi * -0.005555555555555556), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e17

   1. Initial program 65.3%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. difference-of-squares 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. add-cbrt-cube 36.5%

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

      2. pow3 35.2%

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

      3. div-inv 35.2%

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

      4. metadata-eval 35.2%

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

   8. Applied egg-rr 35.2%

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

      1. expm1-log1p-u 25.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt[3]{{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{3}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-undefine 21.4%

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

   10. Applied egg-rr 35.5%

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

       1. expm1-define 51.5%

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

       2. associate-*r* 47.7%

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

       3. *-commutative 47.7%

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

       4. associate-*l* 47.7%

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

       5. *-commutative 47.7%

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

       6. *-commutative 47.7%

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

       7. +-commutative 47.7%

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

       8. associate-*r* 50.6%

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

       9. *-commutative 50.6%

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

   12. Simplified 50.6%

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

       1. expm1-log1p-u 68.0%

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

       2. associate-*r* 68.0%

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

       3. *-commutative 68.0%

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

       4. associate-*l* 82.0%

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

       5. +-commutative 82.0%

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

       6. *-commutative 82.0%

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

   14. Applied egg-rr 82.0%

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

   if 2e17 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000014e246

   1. Initial program 20.9%

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

   3. Simplified 20.7%

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

      1. unpow2 20.7%

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

      2. unpow2 20.7%

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

      3. difference-of-squares 29.8%

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

   6. Applied egg-rr 29.8%

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

   7. Taylor expanded in b around 0 26.0%

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

      1. fma-define 38.1%

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

      2. +-commutative 38.1%

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

      3. *-commutative 38.1%

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

      4. distribute-rgt1-in 38.1%

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

      5. metadata-eval 38.1%

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

      6. mul0-lft 38.1%

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

      7. associate-*r* 38.1%

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

      8. distribute-rgt-out 38.1%

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

      9. *-commutative 38.1%

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

      10. associate-*l* 35.3%

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

      11. mul-1-neg 35.3%

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

   9. Simplified 29.8%

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

       1. add-sqr-sqrt 36.0%

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

       2. pow2 36.0%

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

   11. Applied egg-rr 36.0%

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

   12. Taylor expanded in angle around 0 50.9%

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

   if 2.00000000000000014e246 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.2%

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

   3. Simplified 25.3%

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

      1. unpow2 25.3%

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

      2. unpow2 25.3%

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

      3. difference-of-squares 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. log1p-expm1-u 30.9%

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

      2. log1p-undefine 42.0%

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

      3. div-inv 42.0%

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

      4. metadata-eval 42.0%

         \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot \color{blue}{-0.005555555555555556}\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]

   8. Applied egg-rr 42.0%

      \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, \sin \left(angle \cdot \left(-0.005555555555555556 \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right) \cdot \left(-b\right), \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot {a}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot \left(\left(a - b\right) \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\\ t_1 := {b\_m}^{2} - {a}^{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+241}:\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(\frac{angle\_m}{\frac{-180}{\pi}}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right) + angle\_m \cdot \left(\pi \cdot b\_m\right)\right) - {a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) (- b_m a))) (t_1 (- (pow b_m 2.0) (pow a 2.0))))
   (*
    angle_s
    (if (<= t_1 (- INFINITY))
      (* 0.011111111111111112 (* a t_0))
      (if (<= t_1 1e+241)
        (*
         (* 2.0 (* (* (- a b_m) (+ a b_m)) (sin (* angle_m (/ PI -180.0)))))
         (cos (/ angle_m (/ -180.0 PI))))
        (if (<= t_1 INFINITY)
          (*
           0.011111111111111112
           (-
            (* b_m (+ (* angle_m (* PI (- a a))) (* angle_m (* PI b_m))))
            (* (pow a 2.0) (* angle_m PI))))
          (* t_0 (* a 0.011111111111111112))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * (b_m - a);
	double t_1 = pow(b_m, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.011111111111111112 * (a * t_0);
	} else if (t_1 <= 1e+241) {
		tmp = (2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (((double) M_PI) / -180.0))))) * cos((angle_m / (-180.0 / ((double) M_PI))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (((double) M_PI) * (a - a))) + (angle_m * (((double) M_PI) * b_m)))) - (pow(a, 2.0) * (angle_m * ((double) M_PI))));
	} else {
		tmp = t_0 * (a * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (angle_m * Math.PI) * (b_m - a);
	double t_1 = Math.pow(b_m, 2.0) - Math.pow(a, 2.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.011111111111111112 * (a * t_0);
	} else if (t_1 <= 1e+241) {
		tmp = (2.0 * (((a - b_m) * (a + b_m)) * Math.sin((angle_m * (Math.PI / -180.0))))) * Math.cos((angle_m / (-180.0 / Math.PI)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (Math.PI * (a - a))) + (angle_m * (Math.PI * b_m)))) - (Math.pow(a, 2.0) * (angle_m * Math.PI)));
	} else {
		tmp = t_0 * (a * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (angle_m * math.pi) * (b_m - a)
	t_1 = math.pow(b_m, 2.0) - math.pow(a, 2.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 0.011111111111111112 * (a * t_0)
	elif t_1 <= 1e+241:
		tmp = (2.0 * (((a - b_m) * (a + b_m)) * math.sin((angle_m * (math.pi / -180.0))))) * math.cos((angle_m / (-180.0 / math.pi)))
	elif t_1 <= math.inf:
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (math.pi * (a - a))) + (angle_m * (math.pi * b_m)))) - (math.pow(a, 2.0) * (angle_m * math.pi)))
	else:
		tmp = t_0 * (a * 0.011111111111111112)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * Float64(b_m - a))
	t_1 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.011111111111111112 * Float64(a * t_0));
	elseif (t_1 <= 1e+241)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(a - b_m) * Float64(a + b_m)) * sin(Float64(angle_m * Float64(pi / -180.0))))) * cos(Float64(angle_m / Float64(-180.0 / pi))));
	elseif (t_1 <= Inf)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(Float64(angle_m * Float64(pi * Float64(a - a))) + Float64(angle_m * Float64(pi * b_m)))) - Float64((a ^ 2.0) * Float64(angle_m * pi))));
	else
		tmp = Float64(t_0 * Float64(a * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (angle_m * pi) * (b_m - a);
	t_1 = (b_m ^ 2.0) - (a ^ 2.0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 0.011111111111111112 * (a * t_0);
	elseif (t_1 <= 1e+241)
		tmp = (2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (pi / -180.0))))) * cos((angle_m / (-180.0 / pi)));
	elseif (t_1 <= Inf)
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (pi * (a - a))) + (angle_m * (pi * b_m)))) - ((a ^ 2.0) * (angle_m * pi)));
	else
		tmp = t_0 * (a * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.011111111111111112 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+241], N[(N[(2.0 * N[(N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m / N[(-180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(0.011111111111111112 * N[(N[(b$95$m * N[(N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(angle$95$m * N[(Pi * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(a * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 63.1%

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

   3. Taylor expanded in angle around 0 61.1%

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

      1. unpow2 61.1%

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

      2. unpow2 61.1%

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

      3. difference-of-squares 61.1%

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

   5. Applied egg-rr 61.1%

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

   6. Taylor expanded in b around 0 61.1%

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

   7. Taylor expanded in angle around 0 82.5%

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

      1. associate-*r* 82.6%

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

   9. Simplified 82.6%

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

   if -inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 1.0000000000000001e241

   1. Initial program 66.5%

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

   3. Simplified 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. difference-of-squares 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. clear-num 66.8%

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

      2. un-div-inv 67.0%

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

   8. Applied egg-rr 67.0%

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

   if 1.0000000000000001e241 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 44.6%

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

   3. Taylor expanded in angle around 0 49.6%

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

      1. unpow2 49.6%

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

      2. unpow2 49.6%

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

      3. difference-of-squares 49.6%

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

   5. Applied egg-rr 49.6%

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

   6. Taylor expanded in b around 0 71.9%

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

   if +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. difference-of-squares 64.4%

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

   5. Applied egg-rr 64.4%

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

   6. Taylor expanded in b around 0 38.9%

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

   7. Taylor expanded in angle around 0 49.9%

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

      1. associate-*r* 49.9%

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

      2. associate-*r* 50.0%

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

   9. Simplified 50.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -\infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 10^{+241}:\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(\frac{angle}{\frac{-180}{\pi}}\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right) + angle \cdot \left(\pi \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\\ t_1 := {b\_m}^{2} - {a}^{2}\\ t_2 := angle\_m \cdot \frac{\pi}{-180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+241}:\\ \;\;\;\;\cos t\_2 \cdot \left(2 \cdot \left(\left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right) \cdot \sin t\_2\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right) + angle\_m \cdot \left(\pi \cdot b\_m\right)\right) - {a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) (- b_m a)))
        (t_1 (- (pow b_m 2.0) (pow a 2.0)))
        (t_2 (* angle_m (/ PI -180.0))))
   (*
    angle_s
    (if (<= t_1 (- INFINITY))
      (* 0.011111111111111112 (* a t_0))
      (if (<= t_1 1e+241)
        (* (cos t_2) (* 2.0 (* (* (- a b_m) (+ a b_m)) (sin t_2))))
        (if (<= t_1 INFINITY)
          (*
           0.011111111111111112
           (-
            (* b_m (+ (* angle_m (* PI (- a a))) (* angle_m (* PI b_m))))
            (* (pow a 2.0) (* angle_m PI))))
          (* t_0 (* a 0.011111111111111112))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * (b_m - a);
	double t_1 = pow(b_m, 2.0) - pow(a, 2.0);
	double t_2 = angle_m * (((double) M_PI) / -180.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.011111111111111112 * (a * t_0);
	} else if (t_1 <= 1e+241) {
		tmp = cos(t_2) * (2.0 * (((a - b_m) * (a + b_m)) * sin(t_2)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (((double) M_PI) * (a - a))) + (angle_m * (((double) M_PI) * b_m)))) - (pow(a, 2.0) * (angle_m * ((double) M_PI))));
	} else {
		tmp = t_0 * (a * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (angle_m * Math.PI) * (b_m - a);
	double t_1 = Math.pow(b_m, 2.0) - Math.pow(a, 2.0);
	double t_2 = angle_m * (Math.PI / -180.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.011111111111111112 * (a * t_0);
	} else if (t_1 <= 1e+241) {
		tmp = Math.cos(t_2) * (2.0 * (((a - b_m) * (a + b_m)) * Math.sin(t_2)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (Math.PI * (a - a))) + (angle_m * (Math.PI * b_m)))) - (Math.pow(a, 2.0) * (angle_m * Math.PI)));
	} else {
		tmp = t_0 * (a * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (angle_m * math.pi) * (b_m - a)
	t_1 = math.pow(b_m, 2.0) - math.pow(a, 2.0)
	t_2 = angle_m * (math.pi / -180.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 0.011111111111111112 * (a * t_0)
	elif t_1 <= 1e+241:
		tmp = math.cos(t_2) * (2.0 * (((a - b_m) * (a + b_m)) * math.sin(t_2)))
	elif t_1 <= math.inf:
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (math.pi * (a - a))) + (angle_m * (math.pi * b_m)))) - (math.pow(a, 2.0) * (angle_m * math.pi)))
	else:
		tmp = t_0 * (a * 0.011111111111111112)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * Float64(b_m - a))
	t_1 = Float64((b_m ^ 2.0) - (a ^ 2.0))
	t_2 = Float64(angle_m * Float64(pi / -180.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.011111111111111112 * Float64(a * t_0));
	elseif (t_1 <= 1e+241)
		tmp = Float64(cos(t_2) * Float64(2.0 * Float64(Float64(Float64(a - b_m) * Float64(a + b_m)) * sin(t_2))));
	elseif (t_1 <= Inf)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(Float64(angle_m * Float64(pi * Float64(a - a))) + Float64(angle_m * Float64(pi * b_m)))) - Float64((a ^ 2.0) * Float64(angle_m * pi))));
	else
		tmp = Float64(t_0 * Float64(a * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (angle_m * pi) * (b_m - a);
	t_1 = (b_m ^ 2.0) - (a ^ 2.0);
	t_2 = angle_m * (pi / -180.0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 0.011111111111111112 * (a * t_0);
	elseif (t_1 <= 1e+241)
		tmp = cos(t_2) * (2.0 * (((a - b_m) * (a + b_m)) * sin(t_2)));
	elseif (t_1 <= Inf)
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (pi * (a - a))) + (angle_m * (pi * b_m)))) - ((a ^ 2.0) * (angle_m * pi)));
	else
		tmp = t_0 * (a * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.011111111111111112 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+241], N[(N[Cos[t$95$2], $MachinePrecision] * N[(2.0 * N[(N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(0.011111111111111112 * N[(N[(b$95$m * N[(N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(angle$95$m * N[(Pi * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(a * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 63.1%

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

   3. Taylor expanded in angle around 0 61.1%

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

      1. unpow2 61.1%

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

      2. unpow2 61.1%

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

      3. difference-of-squares 61.1%

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

   5. Applied egg-rr 61.1%

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

   6. Taylor expanded in b around 0 61.1%

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

   7. Taylor expanded in angle around 0 82.5%

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

      1. associate-*r* 82.6%

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

   9. Simplified 82.6%

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

   if -inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 1.0000000000000001e241

   1. Initial program 66.5%

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

   3. Simplified 66.8%

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

      1. unpow2 66.8%

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

      2. unpow2 66.8%

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

      3. difference-of-squares 66.8%

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

   6. Applied egg-rr 66.8%

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

   if 1.0000000000000001e241 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 44.6%

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

   3. Taylor expanded in angle around 0 49.6%

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

      1. unpow2 49.6%

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

      2. unpow2 49.6%

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

      3. difference-of-squares 49.6%

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

   5. Applied egg-rr 49.6%

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

   6. Taylor expanded in b around 0 71.9%

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

   if +inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. difference-of-squares 64.4%

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

   5. Applied egg-rr 64.4%

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

   6. Taylor expanded in b around 0 38.9%

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

   7. Taylor expanded in angle around 0 49.9%

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

      1. associate-*r* 49.9%

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

      2. associate-*r* 50.0%

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

   9. Simplified 50.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -\infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 10^{+241}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right) + angle \cdot \left(\pi \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(a \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a - b\_m\right) \cdot \left(a + b\_m\right)\\ t_1 := angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\left(\cos t\_1 \cdot 2\right) \cdot \left(\left(a - b\_m\right) \cdot \left(\left(a + b\_m\right) \cdot \sin t\_1\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(2 \cdot \left(t\_0 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle\_m \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot \left(2 \cdot \left(t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- a b_m) (+ a b_m)))
        (t_1 (* angle_m (* PI -0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+247)
      (* (* (cos t_1) 2.0) (* (- a b_m) (* (+ a b_m) (sin t_1))))
      (if (<= (/ angle_m 180.0) 4e+291)
        (*
         (cos (* angle_m (/ (cbrt (pow PI 3.0)) -180.0)))
         (* 2.0 (* t_0 (sin (/ PI (/ 180.0 angle_m))))))
        (*
         (cos (* angle_m (log (+ 1.0 (expm1 (* PI -0.005555555555555556))))))
         (* 2.0 (* t_0 (sin (* angle_m (/ PI -180.0)))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (a - b_m) * (a + b_m);
	double t_1 = angle_m * (((double) M_PI) * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 5e+247) {
		tmp = (cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * sin(t_1)));
	} else if ((angle_m / 180.0) <= 4e+291) {
		tmp = cos((angle_m * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))) * (2.0 * (t_0 * sin((((double) M_PI) / (180.0 / angle_m)))));
	} else {
		tmp = cos((angle_m * log((1.0 + expm1((((double) M_PI) * -0.005555555555555556)))))) * (2.0 * (t_0 * sin((angle_m * (((double) M_PI) / -180.0)))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (a - b_m) * (a + b_m);
	double t_1 = angle_m * (Math.PI * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 5e+247) {
		tmp = (Math.cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * Math.sin(t_1)));
	} else if ((angle_m / 180.0) <= 4e+291) {
		tmp = Math.cos((angle_m * (Math.cbrt(Math.pow(Math.PI, 3.0)) / -180.0))) * (2.0 * (t_0 * Math.sin((Math.PI / (180.0 / angle_m)))));
	} else {
		tmp = Math.cos((angle_m * Math.log((1.0 + Math.expm1((Math.PI * -0.005555555555555556)))))) * (2.0 * (t_0 * Math.sin((angle_m * (Math.PI / -180.0)))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(a - b_m) * Float64(a + b_m))
	t_1 = Float64(angle_m * Float64(pi * -0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+247)
		tmp = Float64(Float64(cos(t_1) * 2.0) * Float64(Float64(a - b_m) * Float64(Float64(a + b_m) * sin(t_1))));
	elseif (Float64(angle_m / 180.0) <= 4e+291)
		tmp = Float64(cos(Float64(angle_m * Float64(cbrt((pi ^ 3.0)) / -180.0))) * Float64(2.0 * Float64(t_0 * sin(Float64(pi / Float64(180.0 / angle_m))))));
	else
		tmp = Float64(cos(Float64(angle_m * log(Float64(1.0 + expm1(Float64(pi * -0.005555555555555556)))))) * Float64(2.0 * Float64(t_0 * sin(Float64(angle_m * Float64(pi / -180.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+247], N[(N[(N[Cos[t$95$1], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a - b$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+291], N[(N[Cos[N[(angle$95$m * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle$95$m * N[Log[N[(1.0 + N[(Exp[N[(Pi * -0.005555555555555556), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000023e247

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. difference-of-squares 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. add-cbrt-cube 34.4%

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

      2. pow3 32.4%

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

      3. div-inv 32.4%

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

      4. metadata-eval 32.4%

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

   8. Applied egg-rr 32.4%

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

      1. expm1-log1p-u 23.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt[3]{{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{3}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-undefine 20.2%

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

   10. Applied egg-rr 33.3%

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

       1. expm1-define 47.4%

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

       2. associate-*r* 44.0%

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

       3. *-commutative 44.0%

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

       4. associate-*l* 44.0%

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

       5. *-commutative 44.0%

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

       6. *-commutative 44.0%

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

       7. +-commutative 44.0%

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

       8. associate-*r* 45.8%

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

       9. *-commutative 45.8%

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

   12. Simplified 45.8%

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

       1. expm1-log1p-u 62.7%

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

       2. associate-*r* 62.7%

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

       3. *-commutative 62.7%

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

       4. associate-*l* 74.8%

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

       5. +-commutative 74.8%

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

       6. *-commutative 74.8%

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

   14. Applied egg-rr 74.8%

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

   if 5.00000000000000023e247 < (/.f64 angle #s(literal 180 binary64)) < 3.9999999999999998e291

   1. Initial program 10.4%

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

   3. Simplified 10.6%

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

      1. unpow2 10.6%

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

      2. unpow2 10.6%

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

      3. difference-of-squares 19.7%

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

   6. Applied egg-rr 19.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*r/ 0.0%

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

      5. frac-times 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. metadata-eval 0.0%

         \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]

      9. metadata-eval 0.0%

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

      10. frac-times 0.0%

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

      11. associate-*r/ 0.0%

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

      12. associate-*r/ 0.0%

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

      13. sqrt-unprod 20.5%

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

      14. add-sqr-sqrt 14.3%

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

      15. clear-num 14.3%

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

      16. un-div-inv 15.4%

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

   8. Applied egg-rr 15.4%

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

      1. add-cbrt-cube 27.8%

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

      2. pow3 27.8%

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

   10. Applied egg-rr 27.8%

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

   if 3.9999999999999998e291 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

      1. unpow2 48.4%

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

      2. unpow2 48.4%

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

      3. difference-of-squares 48.4%

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

   6. Applied egg-rr 48.4%

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

      1. log1p-expm1-u 48.4%

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

      2. log1p-undefine 74.3%

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

      3. div-inv 74.3%

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

      4. metadata-eval 74.3%

         \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot \color{blue}{-0.005555555555555556}\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]

   8. Applied egg-rr 77.5%

      \[\leadsto \cos \left(angle \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot \left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a - b\_m\right) \cdot \left(a + b\_m\right)\\ t_1 := angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\left(\cos t\_1 \cdot 2\right) \cdot \left(\left(a - b\_m\right) \cdot \left(\left(a + b\_m\right) \cdot \sin t\_1\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+293}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(2 \cdot \left(t\_0 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle\_m \cdot -0.005555555555555556\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- a b_m) (+ a b_m)))
        (t_1 (* angle_m (* PI -0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+247)
      (* (* (cos t_1) 2.0) (* (- a b_m) (* (+ a b_m) (sin t_1))))
      (if (<= (/ angle_m 180.0) 1e+293)
        (*
         (cos (* angle_m (/ (cbrt (pow PI 3.0)) -180.0)))
         (* 2.0 (* t_0 (sin (/ PI (/ 180.0 angle_m))))))
        (*
         (* 2.0 (* t_0 (sin (* angle_m (/ PI -180.0)))))
         (cos (* PI (* angle_m -0.005555555555555556)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (a - b_m) * (a + b_m);
	double t_1 = angle_m * (((double) M_PI) * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 5e+247) {
		tmp = (cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * sin(t_1)));
	} else if ((angle_m / 180.0) <= 1e+293) {
		tmp = cos((angle_m * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))) * (2.0 * (t_0 * sin((((double) M_PI) / (180.0 / angle_m)))));
	} else {
		tmp = (2.0 * (t_0 * sin((angle_m * (((double) M_PI) / -180.0))))) * cos((((double) M_PI) * (angle_m * -0.005555555555555556)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (a - b_m) * (a + b_m);
	double t_1 = angle_m * (Math.PI * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 5e+247) {
		tmp = (Math.cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * Math.sin(t_1)));
	} else if ((angle_m / 180.0) <= 1e+293) {
		tmp = Math.cos((angle_m * (Math.cbrt(Math.pow(Math.PI, 3.0)) / -180.0))) * (2.0 * (t_0 * Math.sin((Math.PI / (180.0 / angle_m)))));
	} else {
		tmp = (2.0 * (t_0 * Math.sin((angle_m * (Math.PI / -180.0))))) * Math.cos((Math.PI * (angle_m * -0.005555555555555556)));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(a - b_m) * Float64(a + b_m))
	t_1 = Float64(angle_m * Float64(pi * -0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+247)
		tmp = Float64(Float64(cos(t_1) * 2.0) * Float64(Float64(a - b_m) * Float64(Float64(a + b_m) * sin(t_1))));
	elseif (Float64(angle_m / 180.0) <= 1e+293)
		tmp = Float64(cos(Float64(angle_m * Float64(cbrt((pi ^ 3.0)) / -180.0))) * Float64(2.0 * Float64(t_0 * sin(Float64(pi / Float64(180.0 / angle_m))))));
	else
		tmp = Float64(Float64(2.0 * Float64(t_0 * sin(Float64(angle_m * Float64(pi / -180.0))))) * cos(Float64(pi * Float64(angle_m * -0.005555555555555556))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+247], N[(N[(N[Cos[t$95$1], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a - b$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+293], N[(N[Cos[N[(angle$95$m * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000023e247

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. difference-of-squares 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. add-cbrt-cube 34.4%

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

      2. pow3 32.4%

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

      3. div-inv 32.4%

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

      4. metadata-eval 32.4%

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

   8. Applied egg-rr 32.4%

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

      1. expm1-log1p-u 23.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt[3]{{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{3}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-undefine 20.2%

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

   10. Applied egg-rr 33.3%

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

       1. expm1-define 47.4%

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

       2. associate-*r* 44.0%

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

       3. *-commutative 44.0%

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

       4. associate-*l* 44.0%

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

       5. *-commutative 44.0%

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

       6. *-commutative 44.0%

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

       7. +-commutative 44.0%

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

       8. associate-*r* 45.8%

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

       9. *-commutative 45.8%

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

   12. Simplified 45.8%

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

       1. expm1-log1p-u 62.7%

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

       2. associate-*r* 62.7%

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

       3. *-commutative 62.7%

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

       4. associate-*l* 74.8%

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

       5. +-commutative 74.8%

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

       6. *-commutative 74.8%

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

   14. Applied egg-rr 74.8%

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

   if 5.00000000000000023e247 < (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999992e292

   1. Initial program 10.8%

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

   3. Simplified 11.0%

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

      1. unpow2 11.0%

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

      2. unpow2 11.0%

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

      3. difference-of-squares 19.4%

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

   6. Applied egg-rr 19.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*r/ 0.0%

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

      5. frac-times 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. metadata-eval 0.0%

         \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]

      9. metadata-eval 0.0%

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

      10. frac-times 0.0%

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

      11. associate-*r/ 0.0%

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

      12. associate-*r/ 0.0%

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

      13. sqrt-unprod 20.5%

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

      14. add-sqr-sqrt 13.4%

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

      15. clear-num 13.4%

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

      16. un-div-inv 14.4%

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

   8. Applied egg-rr 14.4%

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

      1. add-cbrt-cube 26.8%

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

      2. pow3 26.8%

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

   10. Applied egg-rr 26.8%

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

   if 9.9999999999999992e292 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 53.8%

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

   3. Simplified 53.8%

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

      1. unpow2 53.8%

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

      2. unpow2 53.8%

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

      3. difference-of-squares 53.8%

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in angle around inf 69.8%

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

      1. associate-*r* 87.2%

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

   9. Simplified 87.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+293}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;\left(\cos t\_0 \cdot 2\right) \cdot \left(\left(a - b\_m\right) \cdot \left(\left(a + b\_m\right) \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle\_m \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot \left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI -0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+246)
      (* (* (cos t_0) 2.0) (* (- a b_m) (* (+ a b_m) (sin t_0))))
      (*
       (cos (* angle_m (/ PI -180.0)))
       (*
        2.0
        (*
         (sin (* angle_m (log (+ 1.0 (expm1 (* PI -0.005555555555555556))))))
         (* (- a b_m) (+ a b_m)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+246) {
		tmp = (cos(t_0) * 2.0) * ((a - b_m) * ((a + b_m) * sin(t_0)));
	} else {
		tmp = cos((angle_m * (((double) M_PI) / -180.0))) * (2.0 * (sin((angle_m * log((1.0 + expm1((((double) M_PI) * -0.005555555555555556)))))) * ((a - b_m) * (a + b_m))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = angle_m * (Math.PI * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+246) {
		tmp = (Math.cos(t_0) * 2.0) * ((a - b_m) * ((a + b_m) * Math.sin(t_0)));
	} else {
		tmp = Math.cos((angle_m * (Math.PI / -180.0))) * (2.0 * (Math.sin((angle_m * Math.log((1.0 + Math.expm1((Math.PI * -0.005555555555555556)))))) * ((a - b_m) * (a + b_m))));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = angle_m * (math.pi * -0.005555555555555556)
	tmp = 0
	if (angle_m / 180.0) <= 2e+246:
		tmp = (math.cos(t_0) * 2.0) * ((a - b_m) * ((a + b_m) * math.sin(t_0)))
	else:
		tmp = math.cos((angle_m * (math.pi / -180.0))) * (2.0 * (math.sin((angle_m * math.log((1.0 + math.expm1((math.pi * -0.005555555555555556)))))) * ((a - b_m) * (a + b_m))))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(angle_m * Float64(pi * -0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+246)
		tmp = Float64(Float64(cos(t_0) * 2.0) * Float64(Float64(a - b_m) * Float64(Float64(a + b_m) * sin(t_0))));
	else
		tmp = Float64(cos(Float64(angle_m * Float64(pi / -180.0))) * Float64(2.0 * Float64(sin(Float64(angle_m * log(Float64(1.0 + expm1(Float64(pi * -0.005555555555555556)))))) * Float64(Float64(a - b_m) * Float64(a + b_m)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+246], N[(N[(N[Cos[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a - b$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[Sin[N[(angle$95$m * N[Log[N[(1.0 + N[(Exp[N[(Pi * -0.005555555555555556), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000014e246

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. difference-of-squares 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. add-cbrt-cube 34.4%

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

      2. pow3 32.4%

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

      3. div-inv 32.4%

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

      4. metadata-eval 32.4%

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

   8. Applied egg-rr 32.4%

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

      1. expm1-log1p-u 23.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt[3]{{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{3}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-undefine 20.2%

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

   10. Applied egg-rr 33.3%

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

       1. expm1-define 47.4%

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

       2. associate-*r* 44.0%

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

       3. *-commutative 44.0%

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

       4. associate-*l* 44.0%

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

       5. *-commutative 44.0%

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

       6. *-commutative 44.0%

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

       7. +-commutative 44.0%

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

       8. associate-*r* 45.8%

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

       9. *-commutative 45.8%

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

   12. Simplified 45.8%

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

       1. expm1-log1p-u 62.7%

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

       2. associate-*r* 62.7%

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

       3. *-commutative 62.7%

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

       4. associate-*l* 74.8%

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

       5. +-commutative 74.8%

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

       6. *-commutative 74.8%

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

   14. Applied egg-rr 74.8%

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

   if 2.00000000000000014e246 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.2%

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

   3. Simplified 25.3%

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

      1. unpow2 25.3%

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

      2. unpow2 25.3%

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

      3. difference-of-squares 30.9%

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

   6. Applied egg-rr 30.9%

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

      1. log1p-expm1-u 30.9%

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

      2. log1p-undefine 42.0%

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

      3. div-inv 42.0%

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

      4. metadata-eval 42.0%

         \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot \color{blue}{-0.005555555555555556}\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]

   8. Applied egg-rr 42.0%

      \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+246}:\\ \;\;\;\;\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot -0.005555555555555556\right)\right)\right) \cdot \left(\left(a - b\right) \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a - b\_m\right) \cdot \left(a + b\_m\right)\\ t_1 := angle\_m \cdot \left(\pi \cdot -0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+17}:\\ \;\;\;\;\left(\cos t\_1 \cdot 2\right) \cdot \left(\left(a - b\_m\right) \cdot \left(\left(a + b\_m\right) \cdot \sin t\_1\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;\left(2 \cdot \left(t\_0 \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(t\_0 \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle\_m \cdot -0.005555555555555556\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- a b_m) (+ a b_m)))
        (t_1 (* angle_m (* PI -0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+17)
      (* (* (cos t_1) 2.0) (* (- a b_m) (* (+ a b_m) (sin t_1))))
      (if (<= (/ angle_m 180.0) 5e+190)
        (*
         (* 2.0 (* t_0 (sin (/ PI (/ 180.0 angle_m)))))
         (cos (* -0.005555555555555556 (* angle_m PI))))
        (*
         (* 2.0 (* t_0 (sin (* angle_m (/ PI -180.0)))))
         (cos (* PI (* angle_m -0.005555555555555556)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (a - b_m) * (a + b_m);
	double t_1 = angle_m * (((double) M_PI) * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+17) {
		tmp = (cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * sin(t_1)));
	} else if ((angle_m / 180.0) <= 5e+190) {
		tmp = (2.0 * (t_0 * sin((((double) M_PI) / (180.0 / angle_m))))) * cos((-0.005555555555555556 * (angle_m * ((double) M_PI))));
	} else {
		tmp = (2.0 * (t_0 * sin((angle_m * (((double) M_PI) / -180.0))))) * cos((((double) M_PI) * (angle_m * -0.005555555555555556)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (a - b_m) * (a + b_m);
	double t_1 = angle_m * (Math.PI * -0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+17) {
		tmp = (Math.cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * Math.sin(t_1)));
	} else if ((angle_m / 180.0) <= 5e+190) {
		tmp = (2.0 * (t_0 * Math.sin((Math.PI / (180.0 / angle_m))))) * Math.cos((-0.005555555555555556 * (angle_m * Math.PI)));
	} else {
		tmp = (2.0 * (t_0 * Math.sin((angle_m * (Math.PI / -180.0))))) * Math.cos((Math.PI * (angle_m * -0.005555555555555556)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (a - b_m) * (a + b_m)
	t_1 = angle_m * (math.pi * -0.005555555555555556)
	tmp = 0
	if (angle_m / 180.0) <= 1e+17:
		tmp = (math.cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * math.sin(t_1)))
	elif (angle_m / 180.0) <= 5e+190:
		tmp = (2.0 * (t_0 * math.sin((math.pi / (180.0 / angle_m))))) * math.cos((-0.005555555555555556 * (angle_m * math.pi)))
	else:
		tmp = (2.0 * (t_0 * math.sin((angle_m * (math.pi / -180.0))))) * math.cos((math.pi * (angle_m * -0.005555555555555556)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(a - b_m) * Float64(a + b_m))
	t_1 = Float64(angle_m * Float64(pi * -0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+17)
		tmp = Float64(Float64(cos(t_1) * 2.0) * Float64(Float64(a - b_m) * Float64(Float64(a + b_m) * sin(t_1))));
	elseif (Float64(angle_m / 180.0) <= 5e+190)
		tmp = Float64(Float64(2.0 * Float64(t_0 * sin(Float64(pi / Float64(180.0 / angle_m))))) * cos(Float64(-0.005555555555555556 * Float64(angle_m * pi))));
	else
		tmp = Float64(Float64(2.0 * Float64(t_0 * sin(Float64(angle_m * Float64(pi / -180.0))))) * cos(Float64(pi * Float64(angle_m * -0.005555555555555556))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (a - b_m) * (a + b_m);
	t_1 = angle_m * (pi * -0.005555555555555556);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+17)
		tmp = (cos(t_1) * 2.0) * ((a - b_m) * ((a + b_m) * sin(t_1)));
	elseif ((angle_m / 180.0) <= 5e+190)
		tmp = (2.0 * (t_0 * sin((pi / (180.0 / angle_m))))) * cos((-0.005555555555555556 * (angle_m * pi)));
	else
		tmp = (2.0 * (t_0 * sin((angle_m * (pi / -180.0))))) * cos((pi * (angle_m * -0.005555555555555556)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+17], N[(N[(N[Cos[t$95$1], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a - b$95$m), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+190], N[(N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$0 * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e17

   1. Initial program 65.6%

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

   3. Simplified 65.7%

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

      1. unpow2 65.7%

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

      2. unpow2 65.7%

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

      3. difference-of-squares 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. add-cbrt-cube 36.6%

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

      2. pow3 35.3%

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

      3. div-inv 35.3%

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

      4. metadata-eval 35.3%

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

   8. Applied egg-rr 35.3%

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

      1. expm1-log1p-u 25.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt[3]{{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{3}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-undefine 21.5%

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

   10. Applied egg-rr 35.6%

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

       1. expm1-define 51.8%

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

       2. associate-*r* 47.9%

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

       3. *-commutative 47.9%

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

       4. associate-*l* 47.9%

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

       5. *-commutative 47.9%

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

       6. *-commutative 47.9%

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

       7. +-commutative 47.9%

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

       8. associate-*r* 50.8%

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

       9. *-commutative 50.8%

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

   12. Simplified 50.8%

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

       1. expm1-log1p-u 68.3%

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

       2. associate-*r* 68.3%

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

       3. *-commutative 68.3%

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

       4. associate-*l* 82.4%

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

       5. +-commutative 82.4%

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

       6. *-commutative 82.4%

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

   14. Applied egg-rr 82.4%

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

   if 1e17 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000036e190

   1. Initial program 19.8%

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

   3. Simplified 19.3%

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

      1. unpow2 19.3%

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

      2. unpow2 19.3%

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

      3. difference-of-squares 27.0%

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

   6. Applied egg-rr 27.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 38.9%

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

      3. associate-*r/ 35.1%

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

      4. associate-*r/ 31.2%

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

      5. frac-times 31.2%

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

      6. *-commutative 31.2%

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

      7. *-commutative 31.2%

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

      8. metadata-eval 31.2%

         \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt{\frac{\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)}{\color{blue}{32400}}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]

      9. metadata-eval 31.2%

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

      10. frac-times 31.2%

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

      11. associate-*r/ 27.4%

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

      12. associate-*r/ 30.5%

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

      13. sqrt-unprod 34.5%

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

      14. add-sqr-sqrt 39.6%

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

      15. clear-num 44.0%

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

      16. un-div-inv 48.0%

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

   8. Applied egg-rr 48.0%

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

   9. Taylor expanded in angle around inf 51.9%

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

   if 5.00000000000000036e190 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.3%

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

   3. Simplified 24.6%

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

      1. unpow2 24.6%

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

      2. unpow2 24.6%

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

      3. difference-of-squares 32.3%

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

   6. Applied egg-rr 32.3%

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

   7. Taylor expanded in angle around inf 35.3%

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

      1. associate-*r* 36.1%

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

   9. Simplified 36.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+17}:\\ \;\;\;\;\left(\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot 2\right) \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.65 \cdot 10^{-226}:\\ \;\;\;\;{b\_m}^{2} \cdot \left(-\sin \left(\pi \cdot \left(angle\_m \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+84}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right) + angle\_m \cdot \left(\pi \cdot b\_m\right)\right) - {a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \left(\left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2.65e-226)
    (* (pow b_m 2.0) (- (sin (* PI (* angle_m -0.011111111111111112)))))
    (if (<= a 1.1e+84)
      (*
       0.011111111111111112
       (-
        (* b_m (+ (* angle_m (* PI (- a a))) (* angle_m (* PI b_m))))
        (* (pow a 2.0) (* angle_m PI))))
      (if (<= a 1.8e+160)
        (* 2.0 (* (* (- a b_m) (+ a b_m)) (sin (* angle_m (/ PI -180.0)))))
        (* 0.011111111111111112 (* a (* (* angle_m PI) (- b_m a)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 2.65e-226) {
		tmp = pow(b_m, 2.0) * -sin((((double) M_PI) * (angle_m * -0.011111111111111112)));
	} else if (a <= 1.1e+84) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (((double) M_PI) * (a - a))) + (angle_m * (((double) M_PI) * b_m)))) - (pow(a, 2.0) * (angle_m * ((double) M_PI))));
	} else if (a <= 1.8e+160) {
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (((double) M_PI) / -180.0))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * ((double) M_PI)) * (b_m - a)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 2.65e-226) {
		tmp = Math.pow(b_m, 2.0) * -Math.sin((Math.PI * (angle_m * -0.011111111111111112)));
	} else if (a <= 1.1e+84) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (Math.PI * (a - a))) + (angle_m * (Math.PI * b_m)))) - (Math.pow(a, 2.0) * (angle_m * Math.PI)));
	} else if (a <= 1.8e+160) {
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * Math.sin((angle_m * (Math.PI / -180.0))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * Math.PI) * (b_m - a)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 2.65e-226:
		tmp = math.pow(b_m, 2.0) * -math.sin((math.pi * (angle_m * -0.011111111111111112)))
	elif a <= 1.1e+84:
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (math.pi * (a - a))) + (angle_m * (math.pi * b_m)))) - (math.pow(a, 2.0) * (angle_m * math.pi)))
	elif a <= 1.8e+160:
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * math.sin((angle_m * (math.pi / -180.0))))
	else:
		tmp = 0.011111111111111112 * (a * ((angle_m * math.pi) * (b_m - a)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 2.65e-226)
		tmp = Float64((b_m ^ 2.0) * Float64(-sin(Float64(pi * Float64(angle_m * -0.011111111111111112)))));
	elseif (a <= 1.1e+84)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(Float64(angle_m * Float64(pi * Float64(a - a))) + Float64(angle_m * Float64(pi * b_m)))) - Float64((a ^ 2.0) * Float64(angle_m * pi))));
	elseif (a <= 1.8e+160)
		tmp = Float64(2.0 * Float64(Float64(Float64(a - b_m) * Float64(a + b_m)) * sin(Float64(angle_m * Float64(pi / -180.0)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(Float64(angle_m * pi) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 2.65e-226)
		tmp = (b_m ^ 2.0) * -sin((pi * (angle_m * -0.011111111111111112)));
	elseif (a <= 1.1e+84)
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (pi * (a - a))) + (angle_m * (pi * b_m)))) - ((a ^ 2.0) * (angle_m * pi)));
	elseif (a <= 1.8e+160)
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (pi / -180.0))));
	else
		tmp = 0.011111111111111112 * (a * ((angle_m * pi) * (b_m - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 2.65e-226], N[(N[Power[b$95$m, 2.0], $MachinePrecision] * (-N[Sin[N[(Pi * N[(angle$95$m * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[a, 1.1e+84], N[(0.011111111111111112 * N[(N[(b$95$m * N[(N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(angle$95$m * N[(Pi * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+160], N[(2.0 * N[(N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if a < 2.6500000000000002e-226

   1. Initial program 56.9%

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

   3. Simplified 56.7%

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

   6. Applied egg-rr 37.1%

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

   7. Taylor expanded in a around 0 38.1%

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

      1. associate-*r* 38.1%

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

      2. mul-1-neg 38.1%

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

      3. associate-*r* 38.2%

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

   9. Simplified 38.2%

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

   if 2.6500000000000002e-226 < a < 1.0999999999999999e84

   1. Initial program 58.9%

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

   3. Taylor expanded in angle around 0 60.1%

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

      1. unpow2 60.1%

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

      2. unpow2 60.1%

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

      3. difference-of-squares 60.1%

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

   5. Applied egg-rr 60.1%

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

   6. Taylor expanded in b around 0 69.5%

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

   if 1.0999999999999999e84 < a < 1.80000000000000011e160

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

      3. difference-of-squares 77.3%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in angle around 0 92.8%

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

   if 1.80000000000000011e160 < a

   1. Initial program 45.5%

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

   3. Taylor expanded in angle around 0 42.1%

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

      1. unpow2 42.1%

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

      2. unpow2 42.1%

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

      3. difference-of-squares 60.1%

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

   5. Applied egg-rr 60.1%

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

   6. Taylor expanded in b around 0 56.6%

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

   7. Taylor expanded in angle around 0 72.3%

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

      1. associate-*r* 72.3%

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

   9. Simplified 72.3%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.65 \cdot 10^{-226}:\\ \;\;\;\;{b}^{2} \cdot \left(-\sin \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+84}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right) + angle \cdot \left(\pi \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{+83}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right) + angle\_m \cdot \left(\pi \cdot b\_m\right)\right) - {a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0
         (* 2.0 (* (* (- a b_m) (+ a b_m)) (sin (* angle_m (/ PI -180.0)))))))
   (*
    angle_s
    (if (<= a 3.3e-226)
      t_0
      (if (<= a 1.95e+83)
        (*
         0.011111111111111112
         (-
          (* b_m (+ (* angle_m (* PI (- a a))) (* angle_m (* PI b_m))))
          (* (pow a 2.0) (* angle_m PI))))
        (if (<= a 1.8e+160)
          t_0
          (* 0.011111111111111112 (* a (* (* angle_m PI) (- b_m a))))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = 2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (((double) M_PI) / -180.0))));
	double tmp;
	if (a <= 3.3e-226) {
		tmp = t_0;
	} else if (a <= 1.95e+83) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (((double) M_PI) * (a - a))) + (angle_m * (((double) M_PI) * b_m)))) - (pow(a, 2.0) * (angle_m * ((double) M_PI))));
	} else if (a <= 1.8e+160) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * ((double) M_PI)) * (b_m - a)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = 2.0 * (((a - b_m) * (a + b_m)) * Math.sin((angle_m * (Math.PI / -180.0))));
	double tmp;
	if (a <= 3.3e-226) {
		tmp = t_0;
	} else if (a <= 1.95e+83) {
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (Math.PI * (a - a))) + (angle_m * (Math.PI * b_m)))) - (Math.pow(a, 2.0) * (angle_m * Math.PI)));
	} else if (a <= 1.8e+160) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * Math.PI) * (b_m - a)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = 2.0 * (((a - b_m) * (a + b_m)) * math.sin((angle_m * (math.pi / -180.0))))
	tmp = 0
	if a <= 3.3e-226:
		tmp = t_0
	elif a <= 1.95e+83:
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (math.pi * (a - a))) + (angle_m * (math.pi * b_m)))) - (math.pow(a, 2.0) * (angle_m * math.pi)))
	elif a <= 1.8e+160:
		tmp = t_0
	else:
		tmp = 0.011111111111111112 * (a * ((angle_m * math.pi) * (b_m - a)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(Float64(a - b_m) * Float64(a + b_m)) * sin(Float64(angle_m * Float64(pi / -180.0)))))
	tmp = 0.0
	if (a <= 3.3e-226)
		tmp = t_0;
	elseif (a <= 1.95e+83)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b_m * Float64(Float64(angle_m * Float64(pi * Float64(a - a))) + Float64(angle_m * Float64(pi * b_m)))) - Float64((a ^ 2.0) * Float64(angle_m * pi))));
	elseif (a <= 1.8e+160)
		tmp = t_0;
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(Float64(angle_m * pi) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = 2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (pi / -180.0))));
	tmp = 0.0;
	if (a <= 3.3e-226)
		tmp = t_0;
	elseif (a <= 1.95e+83)
		tmp = 0.011111111111111112 * ((b_m * ((angle_m * (pi * (a - a))) + (angle_m * (pi * b_m)))) - ((a ^ 2.0) * (angle_m * pi)));
	elseif (a <= 1.8e+160)
		tmp = t_0;
	else
		tmp = 0.011111111111111112 * (a * ((angle_m * pi) * (b_m - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 3.3e-226], t$95$0, If[LessEqual[a, 1.95e+83], N[(0.011111111111111112 * N[(N[(b$95$m * N[(N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(angle$95$m * N[(Pi * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+160], t$95$0, N[(0.011111111111111112 * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if a < 3.3e-226 or 1.9500000000000001e83 < a < 1.80000000000000011e160

   1. Initial program 58.0%

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

   3. Simplified 57.8%

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

      1. unpow2 57.8%

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

      2. unpow2 57.8%

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

      3. difference-of-squares 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in angle around 0 60.8%

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

   if 3.3e-226 < a < 1.9500000000000001e83

   1. Initial program 58.9%

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

   3. Taylor expanded in angle around 0 60.1%

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

      1. unpow2 60.1%

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

      2. unpow2 60.1%

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

      3. difference-of-squares 60.1%

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

   5. Applied egg-rr 60.1%

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

   6. Taylor expanded in b around 0 69.5%

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

   if 1.80000000000000011e160 < a

   1. Initial program 45.5%

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

   3. Taylor expanded in angle around 0 42.1%

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

      1. unpow2 42.1%

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

      2. unpow2 42.1%

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

      3. difference-of-squares 60.1%

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

   5. Applied egg-rr 60.1%

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

   6. Taylor expanded in b around 0 56.6%

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

   7. Taylor expanded in angle around 0 72.3%

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

      1. associate-*r* 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{-226}:\\ \;\;\;\;2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{+83}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right) + angle \cdot \left(\pi \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+162}:\\ \;\;\;\;2 \cdot \left(\left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 3.3e+162)
    (* 2.0 (* (* (- a b_m) (+ a b_m)) (sin (* angle_m (/ PI -180.0)))))
    (* 0.011111111111111112 (* a (* (* angle_m PI) (- b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 3.3e+162) {
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (((double) M_PI) / -180.0))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * ((double) M_PI)) * (b_m - a)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 3.3e+162) {
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * Math.sin((angle_m * (Math.PI / -180.0))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * Math.PI) * (b_m - a)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 3.3e+162:
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * math.sin((angle_m * (math.pi / -180.0))))
	else:
		tmp = 0.011111111111111112 * (a * ((angle_m * math.pi) * (b_m - a)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 3.3e+162)
		tmp = Float64(2.0 * Float64(Float64(Float64(a - b_m) * Float64(a + b_m)) * sin(Float64(angle_m * Float64(pi / -180.0)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(Float64(angle_m * pi) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 3.3e+162)
		tmp = 2.0 * (((a - b_m) * (a + b_m)) * sin((angle_m * (pi / -180.0))));
	else
		tmp = 0.011111111111111112 * (a * ((angle_m * pi) * (b_m - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 3.3e+162], N[(2.0 * N[(N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 3.29999999999999987e162

   1. Initial program 58.2%

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

   3. Simplified 58.7%

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

      1. unpow2 58.7%

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

      2. unpow2 58.7%

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

      3. difference-of-squares 60.1%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in angle around 0 59.6%

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

   if 3.29999999999999987e162 < a

   1. Initial program 45.5%

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

   3. Taylor expanded in angle around 0 42.1%

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

      1. unpow2 42.1%

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

      2. unpow2 42.1%

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

      3. difference-of-squares 60.1%

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

   5. Applied egg-rr 60.1%

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

   6. Taylor expanded in b around 0 56.6%

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

   7. Taylor expanded in angle around 0 72.3%

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

      1. associate-*r* 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{+162}:\\ \;\;\;\;2 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{+155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 4.2e+155)
    (* 0.011111111111111112 (* angle_m (* PI (* (+ a b_m) (- b_m a)))))
    (* 0.011111111111111112 (* a (* (* angle_m PI) (- b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 4.2e+155) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((a + b_m) * (b_m - a))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * ((double) M_PI)) * (b_m - a)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 4.2e+155) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((a + b_m) * (b_m - a))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * Math.PI) * (b_m - a)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 4.2e+155:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((a + b_m) * (b_m - a))))
	else:
		tmp = 0.011111111111111112 * (a * ((angle_m * math.pi) * (b_m - a)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 4.2e+155)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a + b_m) * Float64(b_m - a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(Float64(angle_m * pi) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 4.2e+155)
		tmp = 0.011111111111111112 * (angle_m * (pi * ((a + b_m) * (b_m - a))));
	else
		tmp = 0.011111111111111112 * (a * ((angle_m * pi) * (b_m - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 4.2e+155], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a + b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 4.2e155

   1. Initial program 58.5%

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

   3. Taylor expanded in angle around 0 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. difference-of-squares 58.7%

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

   5. Applied egg-rr 58.7%

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

   if 4.2e155 < a

   1. Initial program 44.0%

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

   3. Taylor expanded in angle around 0 40.7%

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

      1. unpow2 40.7%

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

      2. unpow2 40.7%

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

      3. difference-of-squares 58.1%

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

   5. Applied egg-rr 58.1%

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

   6. Taylor expanded in b around 0 54.7%

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

   7. Taylor expanded in angle around 0 69.9%

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

      1. associate-*r* 69.9%

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

   9. Simplified 69.9%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{+155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+125}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(a - b\_m\right) \cdot \left(a + b\_m\right)\right) \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.3e+125)
    (* -0.011111111111111112 (* (* (- a b_m) (+ a b_m)) (* angle_m PI)))
    (* 0.011111111111111112 (* a (* (* angle_m PI) (- b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.3e+125) {
		tmp = -0.011111111111111112 * (((a - b_m) * (a + b_m)) * (angle_m * ((double) M_PI)));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * ((double) M_PI)) * (b_m - a)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1.3e+125) {
		tmp = -0.011111111111111112 * (((a - b_m) * (a + b_m)) * (angle_m * Math.PI));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * Math.PI) * (b_m - a)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 1.3e+125:
		tmp = -0.011111111111111112 * (((a - b_m) * (a + b_m)) * (angle_m * math.pi))
	else:
		tmp = 0.011111111111111112 * (a * ((angle_m * math.pi) * (b_m - a)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 1.3e+125)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(Float64(a - b_m) * Float64(a + b_m)) * Float64(angle_m * pi)));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(Float64(angle_m * pi) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 1.3e+125)
		tmp = -0.011111111111111112 * (((a - b_m) * (a + b_m)) * (angle_m * pi));
	else
		tmp = 0.011111111111111112 * (a * ((angle_m * pi) * (b_m - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.3e+125], N[(-0.011111111111111112 * N[(N[(N[(a - b$95$m), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.30000000000000002e125

   1. Initial program 58.0%

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

   3. Simplified 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. difference-of-squares 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. add-cbrt-cube 32.0%

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

      2. pow3 30.8%

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

      3. div-inv 30.8%

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

      4. metadata-eval 30.8%

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

   8. Applied egg-rr 30.8%

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

      1. expm1-log1p-u 24.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\sin \left(\sqrt[3]{{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{3}}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\right)} \]

      2. expm1-undefine 20.8%

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

   10. Applied egg-rr 31.9%

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

       1. expm1-define 46.7%

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

       2. associate-*r* 42.9%

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

       3. *-commutative 42.9%

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

       4. associate-*l* 42.9%

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

       5. *-commutative 42.9%

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

       6. *-commutative 42.9%

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

       7. +-commutative 42.9%

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

       8. associate-*r* 45.3%

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

       9. *-commutative 45.3%

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

   12. Simplified 45.3%

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

   13. Taylor expanded in angle around 0 58.7%

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

       1. associate-*r* 58.7%

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

   15. Simplified 58.7%

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

   if 1.30000000000000002e125 < a

   1. Initial program 49.1%

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

   3. Taylor expanded in angle around 0 43.4%

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

      1. unpow2 43.4%

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

      2. unpow2 43.4%

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

      3. difference-of-squares 58.3%

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

   5. Applied egg-rr 58.3%

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

   6. Taylor expanded in b around 0 55.4%

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

   7. Taylor expanded in angle around 0 68.4%

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

      1. associate-*r* 68.4%

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

   9. Simplified 68.4%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+125}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(a - b\right) \cdot \left(a + b\right)\right) \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.0% accurate, 26.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1160000000:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b\_m \cdot \left(b\_m - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1160000000.0)
    (* 0.011111111111111112 (* angle_m (* PI (* b_m (- b_m a)))))
    (* 0.011111111111111112 (* a (* (* angle_m PI) (- b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1160000000.0) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * (b_m * (b_m - a))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * ((double) M_PI)) * (b_m - a)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 1160000000.0) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * (b_m * (b_m - a))));
	} else {
		tmp = 0.011111111111111112 * (a * ((angle_m * Math.PI) * (b_m - a)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 1160000000.0:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * (b_m * (b_m - a))))
	else:
		tmp = 0.011111111111111112 * (a * ((angle_m * math.pi) * (b_m - a)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 1160000000.0)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b_m * Float64(b_m - a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(Float64(angle_m * pi) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 1160000000.0)
		tmp = 0.011111111111111112 * (angle_m * (pi * (b_m * (b_m - a))));
	else
		tmp = 0.011111111111111112 * (a * ((angle_m * pi) * (b_m - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1160000000.0], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b$95$m * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.16e9

   1. Initial program 55.9%

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

   3. Taylor expanded in angle around 0 54.5%

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

      1. unpow2 54.5%

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

      2. unpow2 54.5%

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

      3. difference-of-squares 56.9%

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

   5. Applied egg-rr 56.9%

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

   6. Taylor expanded in b around inf 41.3%

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

   if 1.16e9 < a

   1. Initial program 60.7%

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

   3. Taylor expanded in angle around 0 55.4%

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

      1. unpow2 55.4%

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

      2. unpow2 55.4%

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

      3. difference-of-squares 66.2%

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

   5. Applied egg-rr 66.2%

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

   6. Taylor expanded in b around 0 55.8%

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

   7. Taylor expanded in angle around 0 65.2%

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

      1. associate-*r* 65.2%

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

   9. Simplified 65.2%

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

Alternative 14: 42.1% accurate, 38.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in angle around 0 54.6%

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

   1. unpow2 54.6%

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

   2. unpow2 54.6%

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

   3. difference-of-squares 58.7%

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

5. Applied egg-rr 58.7%

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

6. Taylor expanded in b around 0 41.7%

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

7. Taylor expanded in angle around 0 46.7%

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

   1. associate-*r* 46.7%

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

9. Simplified 46.7%

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

Alternative 15: 42.1% accurate, 38.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(a * N[(angle$95$m * N[(Pi * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in angle around 0 54.6%

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

   1. unpow2 54.6%

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

   2. unpow2 54.6%

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

   3. difference-of-squares 58.7%

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

5. Applied egg-rr 58.7%

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

6. Taylor expanded in b around 0 41.7%

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

7. Taylor expanded in angle around 0 46.7%

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

Alternative 16: 21.2% accurate, 46.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(a * N[(Pi * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in angle around 0 54.6%

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

   1. unpow2 54.6%

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

   2. unpow2 54.6%

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

   3. difference-of-squares 58.7%

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

5. Applied egg-rr 58.7%

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

6. Taylor expanded in b around 0 41.7%

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

7. Taylor expanded in a around 0 20.1%

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

   1. *-commutative 20.1%

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

9. Simplified 20.1%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(a \cdot \left(\pi \cdot b\right)\right)}\right) \]
10. Add Preprocessing

Alternative 17: 20.2% accurate, 46.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
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$95$m_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(a * N[(angle$95$m * N[(Pi * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in angle around 0 54.6%

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

   1. unpow2 54.6%

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

   2. unpow2 54.6%

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

   3. difference-of-squares 58.7%

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

5. Applied egg-rr 58.7%

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

6. Taylor expanded in b around 0 41.7%

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

7. Taylor expanded in a around 0 20.0%

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

   1. *-commutative 20.0%

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

9. Simplified 20.0%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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)))))