ab-angle->ABCF B

Percentage Accurate: 54.3% → 64.3%

Time: 37.8s

Alternatives: 15

Speedup: 27.9×

• 32.0% 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: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \cos \left(angle \cdot \frac{\pi}{-180}\right)\\ \mathbf{if}\;a\_m \leq 1.01:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \mathsf{fma}\left(b, t\_0 \cdot \left(-b\right), t\_0 \cdot {a\_m}^{2}\right)\right)\\ \mathbf{elif}\;a\_m \leq 1.58 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right)\right)\right) \cdot \cos \left(\frac{angle \cdot \pi}{-180}\right)\\ \mathbf{elif}\;a\_m \leq 1.9 \cdot 10^{+195}:\\ \;\;\;\;2 \cdot \left(-0.005555555555555556 \cdot \left(a\_m \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right) - angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (sin (* -0.005555555555555556 (* angle PI))))
        (t_1 (cos (* angle (/ PI -180.0)))))
   (if (<= a_m 1.01)
     (* t_1 (* 2.0 (fma b (* t_0 (- b)) (* t_0 (pow a_m 2.0)))))
     (if (<= a_m 1.58e+153)
       (*
        (*
         2.0
         (*
          (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))
          (* (+ a_m b) (- a_m b))))
        (cos (/ (* angle PI) -180.0)))
       (if (<= a_m 1.9e+195)
         (*
          2.0
          (*
           -0.005555555555555556
           (- (* a_m (* a_m (* angle PI))) (* angle (* PI (pow b 2.0))))))
         (* t_1 (* 2.0 (* a_m (* a_m t_0)))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = sin((-0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = cos((angle * (((double) M_PI) / -180.0)));
	double tmp;
	if (a_m <= 1.01) {
		tmp = t_1 * (2.0 * fma(b, (t_0 * -b), (t_0 * pow(a_m, 2.0))));
	} else if (a_m <= 1.58e+153) {
		tmp = (2.0 * (sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))) * ((a_m + b) * (a_m - b)))) * cos(((angle * ((double) M_PI)) / -180.0));
	} else if (a_m <= 1.9e+195) {
		tmp = 2.0 * (-0.005555555555555556 * ((a_m * (a_m * (angle * ((double) M_PI)))) - (angle * (((double) M_PI) * pow(b, 2.0)))));
	} else {
		tmp = t_1 * (2.0 * (a_m * (a_m * t_0)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = sin(Float64(-0.005555555555555556 * Float64(angle * pi)))
	t_1 = cos(Float64(angle * Float64(pi / -180.0)))
	tmp = 0.0
	if (a_m <= 1.01)
		tmp = Float64(t_1 * Float64(2.0 * fma(b, Float64(t_0 * Float64(-b)), Float64(t_0 * (a_m ^ 2.0)))));
	elseif (a_m <= 1.58e+153)
		tmp = Float64(Float64(2.0 * Float64(sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0))) * Float64(Float64(a_m + b) * Float64(a_m - b)))) * cos(Float64(Float64(angle * pi) / -180.0)));
	elseif (a_m <= 1.9e+195)
		tmp = Float64(2.0 * Float64(-0.005555555555555556 * Float64(Float64(a_m * Float64(a_m * Float64(angle * pi))) - Float64(angle * Float64(pi * (b ^ 2.0))))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(a_m * Float64(a_m * t_0))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 1.01], N[(t$95$1 * N[(2.0 * N[(b * N[(t$95$0 * (-b)), $MachinePrecision] + N[(t$95$0 * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.58e+153], N[(N[(2.0 * N[(N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.9e+195], N[(2.0 * N[(-0.005555555555555556 * N[(N[(a$95$m * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(angle * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(a$95$m * N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 1.01000000000000001

   1. Initial program 53.2%

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

   3. Simplified 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 57.4%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in b around 0 60.4%

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

      1. fma-define 62.0%

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

      2. +-commutative 62.0%

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

      3. fma-define 62.0%

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

      4. distribute-rgt1-in 62.0%

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

      5. metadata-eval 62.0%

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

      6. mul0-lft 62.0%

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

      7. mul-1-neg 62.0%

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

      8. fmm-def 62.0%

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

      9. *-commutative 62.0%

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

      10. distribute-lft-out-- 62.0%

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

      11. neg-sub0 62.0%

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

      12. *-commutative 62.0%

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

   9. Simplified 62.0%

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

   if 1.01000000000000001 < a < 1.58e153

   1. Initial program 56.2%

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

   3. Simplified 60.2%

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

      1. unpow2 60.2%

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

      2. unpow2 60.2%

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

      3. difference-of-squares 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. add-cbrt-cube 54.6%

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

      2. pow3 54.6%

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

   8. Applied egg-rr 54.6%

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

      1. associate-*r/ 58.8%

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

   10. Applied egg-rr 58.8%

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

   if 1.58e153 < a < 1.9e195

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

      1. unpow2 24.2%

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

      2. unpow2 24.2%

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

      3. difference-of-squares 35.7%

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

   6. Applied egg-rr 35.7%

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

   7. Taylor expanded in angle around 0 46.8%

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

   8. Taylor expanded in angle around 0 46.8%

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

   9. Taylor expanded in a around 0 77.9%

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

       1. +-commutative 77.9%

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

       2. mul-1-neg 77.9%

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

       3. unsub-neg 77.9%

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

       4. +-commutative 77.9%

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

       5. associate-*r* 77.9%

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

       6. distribute-rgt1-in 77.9%

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

       7. metadata-eval 77.9%

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

       8. mul0-lft 77.9%

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

       9. *-commutative 77.9%

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

       10. distribute-lft-out 77.9%

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

   11. Simplified 77.9%

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

   if 1.9e195 < a

   1. Initial program 55.2%

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

   3. Simplified 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 64.3%

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

   6. Applied egg-rr 64.3%

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

      1. add-cbrt-cube 64.3%

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

      2. pow3 64.3%

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

   8. Applied egg-rr 64.3%

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

   9. Taylor expanded in b around 0 64.3%

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

       1. associate-*r* 64.3%

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

       2. fma-define 64.3%

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

       3. distribute-rgt1-in 64.3%

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

       4. metadata-eval 64.3%

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

       5. mul0-lft 64.3%

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

       6. fma-undefine 64.3%

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

   11. Simplified 87.4%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.01:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-b\right), \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}\right)\right)\\ \mathbf{elif}\;a \leq 1.58 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot \cos \left(\frac{angle \cdot \pi}{-180}\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+195}:\\ \;\;\;\;2 \cdot \left(-0.005555555555555556 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) - angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := a\_m \cdot \left(a\_m \cdot t\_0\right)\\ t_2 := \cos \left(angle \cdot \frac{\pi}{-180}\right)\\ \mathbf{if}\;a\_m \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;t\_2 \cdot \left(2 \cdot \mathsf{fma}\left(b, t\_0 \cdot \left(-b\right), t\_0 \cdot {a\_m}^{2}\right)\right)\\ \mathbf{elif}\;a\_m \leq 1.2 \cdot 10^{+192}:\\ \;\;\;\;\cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left(t\_1 - t\_0 \cdot {b}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(2 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (sin (* -0.005555555555555556 (* angle PI))))
        (t_1 (* a_m (* a_m t_0)))
        (t_2 (cos (* angle (/ PI -180.0)))))
   (if (<= a_m 4.4e+43)
     (* t_2 (* 2.0 (fma b (* t_0 (- b)) (* t_0 (pow a_m 2.0)))))
     (if (<= a_m 1.2e+192)
       (*
        (cos (expm1 (log1p (* PI (* angle 0.005555555555555556)))))
        (* 2.0 (- t_1 (* t_0 (pow b 2.0)))))
       (* t_2 (* 2.0 t_1))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = sin((-0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = a_m * (a_m * t_0);
	double t_2 = cos((angle * (((double) M_PI) / -180.0)));
	double tmp;
	if (a_m <= 4.4e+43) {
		tmp = t_2 * (2.0 * fma(b, (t_0 * -b), (t_0 * pow(a_m, 2.0))));
	} else if (a_m <= 1.2e+192) {
		tmp = cos(expm1(log1p((((double) M_PI) * (angle * 0.005555555555555556))))) * (2.0 * (t_1 - (t_0 * pow(b, 2.0))));
	} else {
		tmp = t_2 * (2.0 * t_1);
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = sin(Float64(-0.005555555555555556 * Float64(angle * pi)))
	t_1 = Float64(a_m * Float64(a_m * t_0))
	t_2 = cos(Float64(angle * Float64(pi / -180.0)))
	tmp = 0.0
	if (a_m <= 4.4e+43)
		tmp = Float64(t_2 * Float64(2.0 * fma(b, Float64(t_0 * Float64(-b)), Float64(t_0 * (a_m ^ 2.0)))));
	elseif (a_m <= 1.2e+192)
		tmp = Float64(cos(expm1(log1p(Float64(pi * Float64(angle * 0.005555555555555556))))) * Float64(2.0 * Float64(t_1 - Float64(t_0 * (b ^ 2.0)))));
	else
		tmp = Float64(t_2 * Float64(2.0 * t_1));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(a$95$m * N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 4.4e+43], N[(t$95$2 * N[(2.0 * N[(b * N[(t$95$0 * (-b)), $MachinePrecision] + N[(t$95$0 * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.2e+192], N[(N[Cos[N[(Exp[N[Log[1 + N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(t$95$1 - N[(t$95$0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.40000000000000001e43

   1. Initial program 54.0%

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

   3. Simplified 55.9%

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

      1. unpow2 55.9%

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

      2. unpow2 55.9%

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

      3. difference-of-squares 58.0%

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

   6. Applied egg-rr 58.0%

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

   7. Taylor expanded in b around 0 60.9%

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

      1. fma-define 62.5%

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

      2. +-commutative 62.5%

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

      3. fma-define 62.5%

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

      4. distribute-rgt1-in 62.5%

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

      5. metadata-eval 62.5%

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

      6. mul0-lft 62.5%

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

      7. mul-1-neg 62.5%

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

      8. fmm-def 62.5%

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

      9. *-commutative 62.5%

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

      10. distribute-lft-out-- 62.5%

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

      11. neg-sub0 62.5%

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

      12. *-commutative 62.5%

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

   9. Simplified 62.5%

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

   if 4.40000000000000001e43 < a < 1.1999999999999999e192

   1. Initial program 43.2%

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

   3. Simplified 46.7%

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

      1. unpow2 46.7%

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

      2. unpow2 46.7%

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

      3. difference-of-squares 50.0%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in a around 0 52.7%

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

      1. +-commutative 52.7%

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

      2. mul-1-neg 52.7%

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

      3. unsub-neg 52.7%

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

   9. Simplified 52.7%

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

       1. associate-*r/ 58.6%

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

       2. div-inv 55.3%

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

       3. metadata-eval 55.3%

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

       4. *-commutative 55.3%

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

       5. add-sqr-sqrt 30.9%

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

       6. sqrt-unprod 53.5%

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

       7. swap-sqr 56.8%

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

       8. metadata-eval 56.8%

          \[\leadsto \cos \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) - \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)\right) \]

       9. metadata-eval 56.8%

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

       10. swap-sqr 53.5%

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

       11. sqrt-unprod 30.4%

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

       12. add-sqr-sqrt 55.3%

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

       13. expm1-log1p-u 52.1%

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

       14. associate-*r* 52.1%

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

   11. Applied egg-rr 52.1%

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

   if 1.1999999999999999e192 < a

   1. Initial program 55.0%

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

   3. Simplified 55.0%

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

      1. unpow2 55.0%

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

      2. unpow2 55.0%

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

      3. difference-of-squares 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. add-cbrt-cube 63.4%

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

      2. pow3 63.4%

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

   8. Applied egg-rr 63.4%

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

   9. Taylor expanded in b around 0 63.4%

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

       1. associate-*r* 63.4%

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

       2. fma-define 63.4%

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

       3. distribute-rgt1-in 63.4%

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

       4. metadata-eval 63.4%

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

       5. mul0-lft 63.4%

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

       6. fma-undefine 63.4%

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

   11. Simplified 88.3%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{+43}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-b\right), \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}\right)\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+192}:\\ \;\;\;\;\cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) - \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq 6.2 \cdot 10^{+144}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot t\_0\right) - t\_0 \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+241}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, t\_0 \cdot \left(-b\right), t\_0 \cdot {a\_m}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (sin (* -0.005555555555555556 (* angle PI)))))
   (if (<= b 6.2e+144)
     (*
      (cos (* angle (/ PI -180.0)))
      (* 2.0 (- (* a_m (* a_m t_0)) (* t_0 (pow b 2.0)))))
     (if (<= b 5e+241)
       (* 2.0 (fma b (* t_0 (- b)) (* t_0 (pow a_m 2.0))))
       (*
        (cos (expm1 (log1p (* PI (* angle 0.005555555555555556)))))
        (*
         2.0
         (*
          (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))
          (* (+ a_m b) (- a_m b)))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = sin((-0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b <= 6.2e+144) {
		tmp = cos((angle * (((double) M_PI) / -180.0))) * (2.0 * ((a_m * (a_m * t_0)) - (t_0 * pow(b, 2.0))));
	} else if (b <= 5e+241) {
		tmp = 2.0 * fma(b, (t_0 * -b), (t_0 * pow(a_m, 2.0)));
	} else {
		tmp = cos(expm1(log1p((((double) M_PI) * (angle * 0.005555555555555556))))) * (2.0 * (sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))) * ((a_m + b) * (a_m - b))));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = sin(Float64(-0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b <= 6.2e+144)
		tmp = Float64(cos(Float64(angle * Float64(pi / -180.0))) * Float64(2.0 * Float64(Float64(a_m * Float64(a_m * t_0)) - Float64(t_0 * (b ^ 2.0)))));
	elseif (b <= 5e+241)
		tmp = Float64(2.0 * fma(b, Float64(t_0 * Float64(-b)), Float64(t_0 * (a_m ^ 2.0))));
	else
		tmp = Float64(cos(expm1(log1p(Float64(pi * Float64(angle * 0.005555555555555556))))) * Float64(2.0 * Float64(sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0))) * Float64(Float64(a_m + b) * Float64(a_m - b)))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 6.2e+144], N[(N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(a$95$m * N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+241], N[(2.0 * N[(b * N[(t$95$0 * (-b)), $MachinePrecision] + N[(t$95$0 * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(Exp[N[Log[1 + N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.2000000000000003e144

   1. Initial program 56.5%

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

   3. Simplified 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 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in a around 0 64.6%

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

      1. +-commutative 64.6%

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

      2. mul-1-neg 64.6%

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

      3. unsub-neg 64.6%

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

   9. Simplified 64.6%

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

   if 6.2000000000000003e144 < b < 5.00000000000000025e241

   1. Initial program 23.8%

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

   3. Simplified 27.4%

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

      1. unpow2 27.4%

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

      2. unpow2 27.4%

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

      3. difference-of-squares 35.6%

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

   6. Applied egg-rr 35.6%

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

   7. Taylor expanded in angle around 0 39.8%

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

   8. Taylor expanded in b around 0 59.2%

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

      1. fma-define 58.5%

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

      2. +-commutative 58.5%

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

      3. fma-define 58.5%

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

      4. distribute-rgt1-in 58.5%

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

      5. metadata-eval 58.5%

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

      6. mul0-lft 58.5%

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

      7. mul-1-neg 58.5%

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

      8. fmm-def 58.5%

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

      9. *-commutative 58.5%

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

      10. distribute-lft-out-- 58.5%

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

      11. neg-sub0 58.5%

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

      12. *-commutative 58.5%

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

   10. Simplified 62.7%

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

   if 5.00000000000000025e241 < b

   1. Initial program 54.0%

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

   3. Simplified 61.7%

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

      1. unpow2 61.7%

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

      2. unpow2 61.7%

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

      3. difference-of-squares 69.4%

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

   6. Applied egg-rr 69.4%

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

      1. add-cbrt-cube 77.0%

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

      2. pow3 77.0%

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

   8. Applied egg-rr 77.0%

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

      1. associate-*r/ 69.4%

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

      2. div-inv 69.4%

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

      3. metadata-eval 69.4%

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

      4. *-commutative 69.4%

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

      5. add-sqr-sqrt 46.2%

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

      6. sqrt-unprod 69.4%

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

      7. swap-sqr 69.4%

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

      8. metadata-eval 69.4%

         \[\leadsto \cos \left(\sqrt{\color{blue}{3.08641975308642 \cdot 10^{-5}} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) - \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)\right) \]

      9. metadata-eval 69.4%

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

      10. swap-sqr 69.4%

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

      11. sqrt-unprod 23.2%

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

      12. add-sqr-sqrt 69.4%

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

      13. expm1-log1p-u 61.7%

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

      14. associate-*r* 61.7%

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

   10. Applied egg-rr 61.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+144}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) - \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+241}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-b\right), \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := -0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;a\_m \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, t\_1 \cdot \left(-b\right), t\_1 \cdot {a\_m}^{2}\right)\\ \mathbf{elif}\;a\_m \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right)\right)\right) \cdot \cos t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot t\_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* -0.005555555555555556 (* angle PI))) (t_1 (sin t_0)))
   (if (<= a_m 1.45e-18)
     (* 2.0 (fma b (* t_1 (- b)) (* t_1 (pow a_m 2.0))))
     (if (<= a_m 1.25e+150)
       (*
        (*
         2.0
         (*
          (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))
          (* (+ a_m b) (- a_m b))))
        (cos t_0))
       (* (cos (* angle (/ PI -180.0))) (* 2.0 (* a_m (* a_m t_1))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = -0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double tmp;
	if (a_m <= 1.45e-18) {
		tmp = 2.0 * fma(b, (t_1 * -b), (t_1 * pow(a_m, 2.0)));
	} else if (a_m <= 1.25e+150) {
		tmp = (2.0 * (sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))) * ((a_m + b) * (a_m - b)))) * cos(t_0);
	} else {
		tmp = cos((angle * (((double) M_PI) / -180.0))) * (2.0 * (a_m * (a_m * t_1)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(-0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	tmp = 0.0
	if (a_m <= 1.45e-18)
		tmp = Float64(2.0 * fma(b, Float64(t_1 * Float64(-b)), Float64(t_1 * (a_m ^ 2.0))));
	elseif (a_m <= 1.25e+150)
		tmp = Float64(Float64(2.0 * Float64(sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0))) * Float64(Float64(a_m + b) * Float64(a_m - b)))) * cos(t_0));
	else
		tmp = Float64(cos(Float64(angle * Float64(pi / -180.0))) * Float64(2.0 * Float64(a_m * Float64(a_m * t_1))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 1.45e-18], N[(2.0 * N[(b * N[(t$95$1 * (-b)), $MachinePrecision] + N[(t$95$1 * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.25e+150], N[(N[(2.0 * N[(N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(a$95$m * N[(a$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.45e-18

   1. Initial program 53.3%

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

   3. Simplified 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 57.5%

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

   6. Applied egg-rr 57.5%

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

   7. Taylor expanded in angle around 0 53.7%

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

   8. Taylor expanded in b around 0 56.7%

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

      1. fma-define 62.1%

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

      2. +-commutative 62.1%

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

      3. fma-define 62.1%

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

      4. distribute-rgt1-in 62.1%

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

      5. metadata-eval 62.1%

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

      6. mul0-lft 62.1%

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

      7. mul-1-neg 62.1%

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

      8. fmm-def 62.1%

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

      9. *-commutative 62.1%

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

      10. distribute-lft-out-- 62.1%

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

      11. neg-sub0 62.1%

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

      12. *-commutative 62.1%

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

   10. Simplified 58.9%

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

   if 1.45e-18 < a < 1.25000000000000002e150

   1. Initial program 55.8%

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

   3. Simplified 59.6%

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

      1. unpow2 59.6%

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

      2. unpow2 59.6%

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

      3. difference-of-squares 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. add-cbrt-cube 54.3%

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

      2. pow3 54.3%

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

   8. Applied egg-rr 54.3%

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

   9. Taylor expanded in angle around inf 57.8%

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

   if 1.25000000000000002e150 < a

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

      1. unpow2 46.7%

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

      2. unpow2 46.7%

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

      3. difference-of-squares 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. add-cbrt-cube 59.5%

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

      2. pow3 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in b around 0 53.4%

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

       1. associate-*r* 53.4%

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

       2. fma-define 53.4%

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

       3. distribute-rgt1-in 53.4%

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

       4. metadata-eval 53.4%

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

       5. mul0-lft 53.4%

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

       6. fma-undefine 53.4%

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

   11. Simplified 78.7%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-b\right), \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, t\_0 \cdot \left(-b\right), t\_0 \cdot {a\_m}^{2}\right)\\ \mathbf{elif}\;a\_m \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right)\right)\right) \cdot \cos \left(\frac{angle \cdot \pi}{-180}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (sin (* -0.005555555555555556 (* angle PI)))))
   (if (<= a_m 1.55e-18)
     (* 2.0 (fma b (* t_0 (- b)) (* t_0 (pow a_m 2.0))))
     (if (<= a_m 1.45e+153)
       (*
        (*
         2.0
         (*
          (sin (* angle (/ (cbrt (pow PI 3.0)) -180.0)))
          (* (+ a_m b) (- a_m b))))
        (cos (/ (* angle PI) -180.0)))
       (* (cos (* angle (/ PI -180.0))) (* 2.0 (* a_m (* a_m t_0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = sin((-0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 1.55e-18) {
		tmp = 2.0 * fma(b, (t_0 * -b), (t_0 * pow(a_m, 2.0)));
	} else if (a_m <= 1.45e+153) {
		tmp = (2.0 * (sin((angle * (cbrt(pow(((double) M_PI), 3.0)) / -180.0))) * ((a_m + b) * (a_m - b)))) * cos(((angle * ((double) M_PI)) / -180.0));
	} else {
		tmp = cos((angle * (((double) M_PI) / -180.0))) * (2.0 * (a_m * (a_m * t_0)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = sin(Float64(-0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 1.55e-18)
		tmp = Float64(2.0 * fma(b, Float64(t_0 * Float64(-b)), Float64(t_0 * (a_m ^ 2.0))));
	elseif (a_m <= 1.45e+153)
		tmp = Float64(Float64(2.0 * Float64(sin(Float64(angle * Float64(cbrt((pi ^ 3.0)) / -180.0))) * Float64(Float64(a_m + b) * Float64(a_m - b)))) * cos(Float64(Float64(angle * pi) / -180.0)));
	else
		tmp = Float64(cos(Float64(angle * Float64(pi / -180.0))) * Float64(2.0 * Float64(a_m * Float64(a_m * t_0))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 1.55e-18], N[(2.0 * N[(b * N[(t$95$0 * (-b)), $MachinePrecision] + N[(t$95$0 * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.45e+153], N[(N[(2.0 * N[(N[Sin[N[(angle * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle * Pi), $MachinePrecision] / -180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(a$95$m * N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.55000000000000003e-18

   1. Initial program 53.3%

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

   3. Simplified 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 57.5%

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

   6. Applied egg-rr 57.5%

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

   7. Taylor expanded in angle around 0 53.7%

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

   8. Taylor expanded in b around 0 56.7%

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

      1. fma-define 62.1%

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

      2. +-commutative 62.1%

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

      3. fma-define 62.1%

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

      4. distribute-rgt1-in 62.1%

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

      5. metadata-eval 62.1%

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

      6. mul0-lft 62.1%

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

      7. mul-1-neg 62.1%

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

      8. fmm-def 62.1%

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

      9. *-commutative 62.1%

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

      10. distribute-lft-out-- 62.1%

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

      11. neg-sub0 62.1%

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

      12. *-commutative 62.1%

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

   10. Simplified 58.9%

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

   if 1.55000000000000003e-18 < a < 1.45000000000000001e153

   1. Initial program 55.8%

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

   3. Simplified 59.6%

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

      1. unpow2 59.6%

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

      2. unpow2 59.6%

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

      3. difference-of-squares 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. add-cbrt-cube 54.3%

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

      2. pow3 54.3%

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

   8. Applied egg-rr 54.3%

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

      1. associate-*r/ 58.2%

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

   10. Applied egg-rr 58.2%

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

   if 1.45000000000000001e153 < a

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

      1. unpow2 46.7%

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

      2. unpow2 46.7%

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

      3. difference-of-squares 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. add-cbrt-cube 59.5%

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

      2. pow3 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in b around 0 53.4%

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

       1. associate-*r* 53.4%

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

       2. fma-define 53.4%

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

       3. distribute-rgt1-in 53.4%

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

       4. metadata-eval 53.4%

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

       5. mul0-lft 53.4%

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

       6. fma-undefine 53.4%

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

   11. Simplified 78.7%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-b\right), \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}\right)\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(angle \cdot \frac{\sqrt[3]{{\pi}^{3}}}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \cdot \cos \left(\frac{angle \cdot \pi}{-180}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 1.9 \cdot 10^{+58}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, t\_0 \cdot \left(-b\right), t\_0 \cdot {a\_m}^{2}\right)\\ \mathbf{elif}\;a\_m \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;{a\_m}^{2} \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{{a\_m}^{2}}, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (sin (* -0.005555555555555556 (* angle PI)))))
   (if (<= a_m 1.9e+58)
     (* 2.0 (fma b (* t_0 (- b)) (* t_0 (pow a_m 2.0))))
     (if (<= a_m 2.6e+150)
       (*
        (pow a_m 2.0)
        (*
         (sin (* angle (* PI 0.011111111111111112)))
         (fma b (/ b (pow a_m 2.0)) -1.0)))
       (* (cos (* angle (/ PI -180.0))) (* 2.0 (* a_m (* a_m t_0))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = sin((-0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 1.9e+58) {
		tmp = 2.0 * fma(b, (t_0 * -b), (t_0 * pow(a_m, 2.0)));
	} else if (a_m <= 2.6e+150) {
		tmp = pow(a_m, 2.0) * (sin((angle * (((double) M_PI) * 0.011111111111111112))) * fma(b, (b / pow(a_m, 2.0)), -1.0));
	} else {
		tmp = cos((angle * (((double) M_PI) / -180.0))) * (2.0 * (a_m * (a_m * t_0)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = sin(Float64(-0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 1.9e+58)
		tmp = Float64(2.0 * fma(b, Float64(t_0 * Float64(-b)), Float64(t_0 * (a_m ^ 2.0))));
	elseif (a_m <= 2.6e+150)
		tmp = Float64((a_m ^ 2.0) * Float64(sin(Float64(angle * Float64(pi * 0.011111111111111112))) * fma(b, Float64(b / (a_m ^ 2.0)), -1.0)));
	else
		tmp = Float64(cos(Float64(angle * Float64(pi / -180.0))) * Float64(2.0 * Float64(a_m * Float64(a_m * t_0))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 1.9e+58], N[(2.0 * N[(b * N[(t$95$0 * (-b)), $MachinePrecision] + N[(t$95$0 * N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 2.6e+150], N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * N[(b / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(a$95$m * N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.8999999999999999e58

   1. Initial program 55.1%

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

   3. Simplified 57.0%

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

      1. unpow2 57.0%

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

      2. unpow2 57.0%

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

      3. difference-of-squares 59.1%

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

   6. Applied egg-rr 59.1%

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

   7. Taylor expanded in angle around 0 55.5%

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

   8. Taylor expanded in b around 0 58.3%

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

      1. fma-define 63.4%

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

      2. +-commutative 63.4%

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

      3. fma-define 63.4%

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

      4. distribute-rgt1-in 63.4%

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

      5. metadata-eval 63.4%

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

      6. mul0-lft 63.4%

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

      7. mul-1-neg 63.4%

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

      8. fmm-def 63.4%

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

      9. *-commutative 63.4%

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

      10. distribute-lft-out-- 63.4%

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

      11. neg-sub0 63.4%

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

      12. *-commutative 63.4%

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

   10. Simplified 60.3%

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

   if 1.8999999999999999e58 < a < 2.60000000000000006e150

   1. Initial program 38.5%

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

      1. associate-*l* 38.5%

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

      2. *-commutative 38.5%

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

      3. associate-*l* 38.5%

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

   3. Simplified 38.5%

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

      1. *-commutative 38.5%

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

      2. flip-- 2.2%

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

      3. associate-*r/ 2.2%

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

   6. Applied egg-rr 1.8%

      \[\leadsto \color{blue}{\frac{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left({b}^{4} - {a}^{4}\right)}{\mathsf{fma}\left(a, a, {b}^{2}\right)}} \]
   7. Step-by-step derivation

      1. associate-/l* 1.8%

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \frac{{b}^{4} - {a}^{4}}{\mathsf{fma}\left(a, a, {b}^{2}\right)}} \]

      2. *-commutative 1.8%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)} \cdot \frac{{b}^{4} - {a}^{4}}{\mathsf{fma}\left(a, a, {b}^{2}\right)} \]

      3. associate-*r* 2.0%

         \[\leadsto \sin \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot 2\right) \cdot \frac{{b}^{4} - {a}^{4}}{\mathsf{fma}\left(a, a, {b}^{2}\right)} \]

      4. *-commutative 2.0%

         \[\leadsto \sin \left(\left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.005555555555555556\right) \cdot 2\right) \cdot \frac{{b}^{4} - {a}^{4}}{\mathsf{fma}\left(a, a, {b}^{2}\right)} \]

      5. associate-*l* 2.0%

         \[\leadsto \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot 2\right)\right)} \cdot \frac{{b}^{4} - {a}^{4}}{\mathsf{fma}\left(a, a, {b}^{2}\right)} \]

      6. metadata-eval 2.0%

         \[\leadsto \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{0.011111111111111112}\right) \cdot \frac{{b}^{4} - {a}^{4}}{\mathsf{fma}\left(a, a, {b}^{2}\right)} \]

   8. Simplified 2.0%

      \[\leadsto \color{blue}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \frac{{b}^{4} - {a}^{4}}{\mathsf{fma}\left(a, a, {b}^{2}\right)}} \]

   9. Taylor expanded in a around inf 42.8%

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

       1. *-commutative 42.8%

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

       2. *-commutative 42.8%

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

       3. associate-/l* 42.8%

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

       4. distribute-lft-out 42.8%

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

       5. *-commutative 42.8%

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

       6. associate-*r* 44.0%

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

       7. +-commutative 44.0%

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

       8. metadata-eval 44.0%

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

       9. sub-neg 44.0%

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

       10. unpow2 44.0%

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

       11. associate-/l* 44.0%

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

       12. fmm-def 44.0%

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

       13. metadata-eval 44.0%

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

   11. Simplified 44.0%

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

   if 2.60000000000000006e150 < a

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

      1. unpow2 46.7%

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

      2. unpow2 46.7%

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

      3. difference-of-squares 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. add-cbrt-cube 59.5%

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

      2. pow3 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in b around 0 53.4%

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

       1. associate-*r* 53.4%

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

       2. fma-define 53.4%

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

       3. distribute-rgt1-in 53.4%

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

       4. metadata-eval 53.4%

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

       5. mul0-lft 53.4%

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

       6. fma-undefine 53.4%

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

   11. Simplified 78.7%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{+58}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(b, \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-b\right), \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {a}^{2}\right)\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;{a}^{2} \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{{a}^{2}}, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a\_m \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\sin t\_0 \cdot \left(\left({b}^{2} - {a\_m}^{2}\right) \cdot \cos t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI 0.005555555555555556))))
   (if (<= a_m 2.6e+150)
     (* 2.0 (* (sin t_0) (* (- (pow b 2.0) (pow a_m 2.0)) (cos t_0))))
     (*
      (cos (* angle (/ PI -180.0)))
      (* 2.0 (* a_m (* a_m (sin (* -0.005555555555555556 (* angle PI))))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * 0.005555555555555556);
	double tmp;
	if (a_m <= 2.6e+150) {
		tmp = 2.0 * (sin(t_0) * ((pow(b, 2.0) - pow(a_m, 2.0)) * cos(t_0)));
	} else {
		tmp = cos((angle * (((double) M_PI) / -180.0))) * (2.0 * (a_m * (a_m * sin((-0.005555555555555556 * (angle * ((double) M_PI)))))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = angle * (Math.PI * 0.005555555555555556);
	double tmp;
	if (a_m <= 2.6e+150) {
		tmp = 2.0 * (Math.sin(t_0) * ((Math.pow(b, 2.0) - Math.pow(a_m, 2.0)) * Math.cos(t_0)));
	} else {
		tmp = Math.cos((angle * (Math.PI / -180.0))) * (2.0 * (a_m * (a_m * Math.sin((-0.005555555555555556 * (angle * Math.PI))))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = angle * (math.pi * 0.005555555555555556)
	tmp = 0
	if a_m <= 2.6e+150:
		tmp = 2.0 * (math.sin(t_0) * ((math.pow(b, 2.0) - math.pow(a_m, 2.0)) * math.cos(t_0)))
	else:
		tmp = math.cos((angle * (math.pi / -180.0))) * (2.0 * (a_m * (a_m * math.sin((-0.005555555555555556 * (angle * math.pi))))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(angle * Float64(pi * 0.005555555555555556))
	tmp = 0.0
	if (a_m <= 2.6e+150)
		tmp = Float64(2.0 * Float64(sin(t_0) * Float64(Float64((b ^ 2.0) - (a_m ^ 2.0)) * cos(t_0))));
	else
		tmp = Float64(cos(Float64(angle * Float64(pi / -180.0))) * Float64(2.0 * Float64(a_m * Float64(a_m * sin(Float64(-0.005555555555555556 * Float64(angle * pi)))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = angle * (pi * 0.005555555555555556);
	tmp = 0.0;
	if (a_m <= 2.6e+150)
		tmp = 2.0 * (sin(t_0) * (((b ^ 2.0) - (a_m ^ 2.0)) * cos(t_0)));
	else
		tmp = cos((angle * (pi / -180.0))) * (2.0 * (a_m * (a_m * sin((-0.005555555555555556 * (angle * pi))))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 2.6e+150], N[(2.0 * N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(a$95$m * N[(a$95$m * N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.60000000000000006e150

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*l* 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in angle around inf 56.2%

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

      1. *-commutative 56.2%

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

      2. *-commutative 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-*r* 54.7%

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

      5. *-commutative 54.7%

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

      6. *-commutative 54.7%

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

      7. associate-*r* 54.0%

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

      8. associate-*l* 54.0%

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

      9. associate-*r* 55.4%

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

      10. *-commutative 55.4%

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

      11. associate-*l* 54.4%

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

   7. Simplified 55.9%

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

   if 2.60000000000000006e150 < a

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

      1. unpow2 46.7%

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

      2. unpow2 46.7%

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

      3. difference-of-squares 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. add-cbrt-cube 59.5%

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

      2. pow3 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in b around 0 53.4%

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

       1. associate-*r* 53.4%

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

       2. fma-define 53.4%

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

       3. distribute-rgt1-in 53.4%

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

       4. metadata-eval 53.4%

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

       5. mul0-lft 53.4%

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

       6. fma-undefine 53.4%

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

   11. Simplified 78.7%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;2 \cdot \left(\left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (pow a_m 2.0) 2e+300)
   (* 2.0 (* (* (+ a_m b) (- a_m b)) (sin (* PI (/ angle -180.0)))))
   (*
    (cos (* angle (/ PI -180.0)))
    (* 2.0 (* a_m (* a_m (sin (* -0.005555555555555556 (* angle PI)))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (pow(a_m, 2.0) <= 2e+300) {
		tmp = 2.0 * (((a_m + b) * (a_m - b)) * sin((((double) M_PI) * (angle / -180.0))));
	} else {
		tmp = cos((angle * (((double) M_PI) / -180.0))) * (2.0 * (a_m * (a_m * sin((-0.005555555555555556 * (angle * ((double) M_PI)))))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if (Math.pow(a_m, 2.0) <= 2e+300) {
		tmp = 2.0 * (((a_m + b) * (a_m - b)) * Math.sin((Math.PI * (angle / -180.0))));
	} else {
		tmp = Math.cos((angle * (Math.PI / -180.0))) * (2.0 * (a_m * (a_m * Math.sin((-0.005555555555555556 * (angle * Math.PI))))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if math.pow(a_m, 2.0) <= 2e+300:
		tmp = 2.0 * (((a_m + b) * (a_m - b)) * math.sin((math.pi * (angle / -180.0))))
	else:
		tmp = math.cos((angle * (math.pi / -180.0))) * (2.0 * (a_m * (a_m * math.sin((-0.005555555555555556 * (angle * math.pi))))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if ((a_m ^ 2.0) <= 2e+300)
		tmp = Float64(2.0 * Float64(Float64(Float64(a_m + b) * Float64(a_m - b)) * sin(Float64(pi * Float64(angle / -180.0)))));
	else
		tmp = Float64(cos(Float64(angle * Float64(pi / -180.0))) * Float64(2.0 * Float64(a_m * Float64(a_m * sin(Float64(-0.005555555555555556 * Float64(angle * pi)))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((a_m ^ 2.0) <= 2e+300)
		tmp = 2.0 * (((a_m + b) * (a_m - b)) * sin((pi * (angle / -180.0))));
	else
		tmp = cos((angle * (pi / -180.0))) * (2.0 * (a_m * (a_m * sin((-0.005555555555555556 * (angle * pi))))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 2e+300], N[(2.0 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(a$95$m * N[(a$95$m * N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 2.0000000000000001e300

   1. Initial program 56.6%

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

   3. Simplified 58.8%

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

      1. unpow2 58.8%

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

      2. unpow2 58.8%

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

      3. difference-of-squares 58.8%

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

   6. Applied egg-rr 58.8%

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

   7. Taylor expanded in angle around 0 54.4%

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

      1. clear-num 54.4%

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

      2. un-div-inv 54.3%

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

   9. Applied egg-rr 54.3%

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

       1. associate-/r/ 56.2%

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

       2. *-commutative 56.2%

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

   11. Simplified 56.2%

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

   if 2.0000000000000001e300 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 42.3%

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

   3. Simplified 43.5%

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

      1. unpow2 43.5%

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

      2. unpow2 43.5%

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

      3. difference-of-squares 54.3%

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

   6. Applied egg-rr 54.3%

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

      1. add-cbrt-cube 51.7%

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

      2. pow3 51.7%

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

   8. Applied egg-rr 51.7%

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

   9. Taylor expanded in b around 0 53.0%

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

       1. associate-*r* 53.0%

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

       2. fma-define 53.0%

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

       3. distribute-rgt1-in 53.0%

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

       4. metadata-eval 53.0%

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

       5. mul0-lft 53.0%

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

       6. fma-undefine 53.0%

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

   11. Simplified 78.4%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := angle \cdot \frac{\pi}{-180}\\ t_1 := -0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;a\_m \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\cos t\_1 \cdot \left(2 \cdot \left(\left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right) \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot \sin t\_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI -180.0)))
        (t_1 (* -0.005555555555555556 (* angle PI))))
   (if (<= a_m 1.25e+150)
     (* (cos t_1) (* 2.0 (* (* (+ a_m b) (- a_m b)) (sin t_0))))
     (* (cos t_0) (* 2.0 (* a_m (* a_m (sin t_1))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = angle * (((double) M_PI) / -180.0);
	double t_1 = -0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (a_m <= 1.25e+150) {
		tmp = cos(t_1) * (2.0 * (((a_m + b) * (a_m - b)) * sin(t_0)));
	} else {
		tmp = cos(t_0) * (2.0 * (a_m * (a_m * sin(t_1))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = angle * (Math.PI / -180.0);
	double t_1 = -0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (a_m <= 1.25e+150) {
		tmp = Math.cos(t_1) * (2.0 * (((a_m + b) * (a_m - b)) * Math.sin(t_0)));
	} else {
		tmp = Math.cos(t_0) * (2.0 * (a_m * (a_m * Math.sin(t_1))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = angle * (math.pi / -180.0)
	t_1 = -0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if a_m <= 1.25e+150:
		tmp = math.cos(t_1) * (2.0 * (((a_m + b) * (a_m - b)) * math.sin(t_0)))
	else:
		tmp = math.cos(t_0) * (2.0 * (a_m * (a_m * math.sin(t_1))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(angle * Float64(pi / -180.0))
	t_1 = Float64(-0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (a_m <= 1.25e+150)
		tmp = Float64(cos(t_1) * Float64(2.0 * Float64(Float64(Float64(a_m + b) * Float64(a_m - b)) * sin(t_0))));
	else
		tmp = Float64(cos(t_0) * Float64(2.0 * Float64(a_m * Float64(a_m * sin(t_1)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = angle * (pi / -180.0);
	t_1 = -0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (a_m <= 1.25e+150)
		tmp = cos(t_1) * (2.0 * (((a_m + b) * (a_m - b)) * sin(t_0)));
	else
		tmp = cos(t_0) * (2.0 * (a_m * (a_m * sin(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 1.25e+150], N[(N[Cos[t$95$1], $MachinePrecision] * N[(2.0 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[t$95$0], $MachinePrecision] * N[(2.0 * N[(a$95$m * N[(a$95$m * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25000000000000002e150

   1. Initial program 53.7%

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

   3. Simplified 55.9%

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

      1. unpow2 55.9%

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

      2. unpow2 55.9%

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

      3. difference-of-squares 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in angle around inf 57.2%

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

   if 1.25000000000000002e150 < a

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

      1. unpow2 46.7%

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

      2. unpow2 46.7%

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

      3. difference-of-squares 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. add-cbrt-cube 59.5%

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

      2. pow3 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in b around 0 53.4%

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

       1. associate-*r* 53.4%

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

       2. fma-define 53.4%

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

       3. distribute-rgt1-in 53.4%

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

       4. metadata-eval 53.4%

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

       5. mul0-lft 53.4%

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

       6. fma-undefine 53.4%

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

   11. Simplified 78.7%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := angle \cdot \frac{\pi}{-180}\\ t_1 := \cos t\_0\\ \mathbf{if}\;a\_m \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right) \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI -180.0))) (t_1 (cos t_0)))
   (if (<= a_m 2.6e+150)
     (* t_1 (* 2.0 (* (* (+ a_m b) (- a_m b)) (sin t_0))))
     (*
      t_1
      (* 2.0 (* a_m (* a_m (sin (* -0.005555555555555556 (* angle PI))))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = angle * (((double) M_PI) / -180.0);
	double t_1 = cos(t_0);
	double tmp;
	if (a_m <= 2.6e+150) {
		tmp = t_1 * (2.0 * (((a_m + b) * (a_m - b)) * sin(t_0)));
	} else {
		tmp = t_1 * (2.0 * (a_m * (a_m * sin((-0.005555555555555556 * (angle * ((double) M_PI)))))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = angle * (Math.PI / -180.0);
	double t_1 = Math.cos(t_0);
	double tmp;
	if (a_m <= 2.6e+150) {
		tmp = t_1 * (2.0 * (((a_m + b) * (a_m - b)) * Math.sin(t_0)));
	} else {
		tmp = t_1 * (2.0 * (a_m * (a_m * Math.sin((-0.005555555555555556 * (angle * Math.PI))))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = angle * (math.pi / -180.0)
	t_1 = math.cos(t_0)
	tmp = 0
	if a_m <= 2.6e+150:
		tmp = t_1 * (2.0 * (((a_m + b) * (a_m - b)) * math.sin(t_0)))
	else:
		tmp = t_1 * (2.0 * (a_m * (a_m * math.sin((-0.005555555555555556 * (angle * math.pi))))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(angle * Float64(pi / -180.0))
	t_1 = cos(t_0)
	tmp = 0.0
	if (a_m <= 2.6e+150)
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(Float64(a_m + b) * Float64(a_m - b)) * sin(t_0))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(a_m * Float64(a_m * sin(Float64(-0.005555555555555556 * Float64(angle * pi)))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = angle * (pi / -180.0);
	t_1 = cos(t_0);
	tmp = 0.0;
	if (a_m <= 2.6e+150)
		tmp = t_1 * (2.0 * (((a_m + b) * (a_m - b)) * sin(t_0)));
	else
		tmp = t_1 * (2.0 * (a_m * (a_m * sin((-0.005555555555555556 * (angle * pi))))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[a$95$m, 2.6e+150], N[(t$95$1 * N[(2.0 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(a$95$m * N[(a$95$m * N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.60000000000000006e150

   1. Initial program 53.7%

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

   3. Simplified 55.9%

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

      1. unpow2 55.9%

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

      2. unpow2 55.9%

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

      3. difference-of-squares 57.8%

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

   6. Applied egg-rr 57.8%

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

   if 2.60000000000000006e150 < a

   1. Initial program 46.7%

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

   3. Simplified 46.7%

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

      1. unpow2 46.7%

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

      2. unpow2 46.7%

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

      3. difference-of-squares 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. add-cbrt-cube 59.5%

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

      2. pow3 59.5%

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

   8. Applied egg-rr 59.5%

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

   9. Taylor expanded in b around 0 53.4%

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

       1. associate-*r* 53.4%

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

       2. fma-define 53.4%

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

       3. distribute-rgt1-in 53.4%

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

       4. metadata-eval 53.4%

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

       5. mul0-lft 53.4%

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

       6. fma-undefine 53.4%

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

   11. Simplified 78.7%

       \[\leadsto \cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \color{blue}{\left(a \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(0 + a\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(a\_m + b\right) \cdot \left(a\_m - b\right)\\ \mathbf{if}\;angle \leq 16000000000000:\\ \;\;\;\;2 \cdot \left(-0.005555555555555556 \cdot \left(a\_m \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right) - angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\right)\\ \mathbf{elif}\;angle \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot t\_0\right)\\ \mathbf{elif}\;angle \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sin \left(\pi \cdot \frac{angle}{-180}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (+ a_m b) (- a_m b))))
   (if (<= angle 16000000000000.0)
     (*
      2.0
      (*
       -0.005555555555555556
       (- (* a_m (* a_m (* angle PI))) (* angle (* PI (pow b 2.0))))))
     (if (<= angle 4.5e+45)
       (* 2.0 (* (sin (* -0.005555555555555556 (* angle PI))) t_0))
       (if (<= angle 5.5e+121)
         (* 2.0 (* t_0 (sin (* PI (* angle 0.005555555555555556)))))
         (* 2.0 (* t_0 (sin (* PI (/ angle -180.0))))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (a_m + b) * (a_m - b);
	double tmp;
	if (angle <= 16000000000000.0) {
		tmp = 2.0 * (-0.005555555555555556 * ((a_m * (a_m * (angle * ((double) M_PI)))) - (angle * (((double) M_PI) * pow(b, 2.0)))));
	} else if (angle <= 4.5e+45) {
		tmp = 2.0 * (sin((-0.005555555555555556 * (angle * ((double) M_PI)))) * t_0);
	} else if (angle <= 5.5e+121) {
		tmp = 2.0 * (t_0 * sin((((double) M_PI) * (angle * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * sin((((double) M_PI) * (angle / -180.0))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = (a_m + b) * (a_m - b);
	double tmp;
	if (angle <= 16000000000000.0) {
		tmp = 2.0 * (-0.005555555555555556 * ((a_m * (a_m * (angle * Math.PI))) - (angle * (Math.PI * Math.pow(b, 2.0)))));
	} else if (angle <= 4.5e+45) {
		tmp = 2.0 * (Math.sin((-0.005555555555555556 * (angle * Math.PI))) * t_0);
	} else if (angle <= 5.5e+121) {
		tmp = 2.0 * (t_0 * Math.sin((Math.PI * (angle * 0.005555555555555556))));
	} else {
		tmp = 2.0 * (t_0 * Math.sin((Math.PI * (angle / -180.0))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = (a_m + b) * (a_m - b)
	tmp = 0
	if angle <= 16000000000000.0:
		tmp = 2.0 * (-0.005555555555555556 * ((a_m * (a_m * (angle * math.pi))) - (angle * (math.pi * math.pow(b, 2.0)))))
	elif angle <= 4.5e+45:
		tmp = 2.0 * (math.sin((-0.005555555555555556 * (angle * math.pi))) * t_0)
	elif angle <= 5.5e+121:
		tmp = 2.0 * (t_0 * math.sin((math.pi * (angle * 0.005555555555555556))))
	else:
		tmp = 2.0 * (t_0 * math.sin((math.pi * (angle / -180.0))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(a_m + b) * Float64(a_m - b))
	tmp = 0.0
	if (angle <= 16000000000000.0)
		tmp = Float64(2.0 * Float64(-0.005555555555555556 * Float64(Float64(a_m * Float64(a_m * Float64(angle * pi))) - Float64(angle * Float64(pi * (b ^ 2.0))))));
	elseif (angle <= 4.5e+45)
		tmp = Float64(2.0 * Float64(sin(Float64(-0.005555555555555556 * Float64(angle * pi))) * t_0));
	elseif (angle <= 5.5e+121)
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(pi * Float64(angle * 0.005555555555555556)))));
	else
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(pi * Float64(angle / -180.0)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = (a_m + b) * (a_m - b);
	tmp = 0.0;
	if (angle <= 16000000000000.0)
		tmp = 2.0 * (-0.005555555555555556 * ((a_m * (a_m * (angle * pi))) - (angle * (pi * (b ^ 2.0)))));
	elseif (angle <= 4.5e+45)
		tmp = 2.0 * (sin((-0.005555555555555556 * (angle * pi))) * t_0);
	elseif (angle <= 5.5e+121)
		tmp = 2.0 * (t_0 * sin((pi * (angle * 0.005555555555555556))));
	else
		tmp = 2.0 * (t_0 * sin((pi * (angle / -180.0))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 16000000000000.0], N[(2.0 * N[(-0.005555555555555556 * N[(N[(a$95$m * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(angle * N[(Pi * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 4.5e+45], N[(2.0 * N[(N[Sin[N[(-0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 5.5e+121], N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 * N[Sin[N[(Pi * N[(angle / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if angle < 1.6e13

   1. Initial program 60.9%

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

   3. Simplified 62.9%

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

      1. unpow2 62.9%

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

      2. unpow2 62.9%

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

      3. difference-of-squares 66.2%

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

   6. Applied egg-rr 66.2%

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

   7. Taylor expanded in angle around 0 58.5%

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

   8. Taylor expanded in angle around 0 59.6%

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

   9. Taylor expanded in a around 0 61.5%

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

       1. +-commutative 61.5%

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

       2. mul-1-neg 61.5%

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

       3. unsub-neg 61.5%

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

       4. +-commutative 61.5%

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

       5. associate-*r* 61.5%

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

       6. distribute-rgt1-in 61.5%

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

       7. metadata-eval 61.5%

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

       8. mul0-lft 61.5%

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

       9. *-commutative 61.5%

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

       10. distribute-lft-out 61.5%

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

   11. Simplified 61.5%

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

   if 1.6e13 < angle < 4.4999999999999998e45

   1. Initial program 0.2%

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

   3. Simplified 3.8%

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

      1. unpow2 3.8%

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

      2. unpow2 3.8%

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

      3. difference-of-squares 23.8%

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

   6. Applied egg-rr 23.8%

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

   7. Taylor expanded in angle around 0 60.2%

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

   8. Taylor expanded in angle around inf 67.8%

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

      1. *-commutative 67.8%

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

   10. Simplified 67.8%

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

   if 4.4999999999999998e45 < angle < 5.4999999999999998e121

   1. Initial program 25.3%

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

   3. Simplified 21.8%

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

      1. unpow2 21.8%

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

      2. unpow2 21.8%

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

      3. difference-of-squares 21.8%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in angle around 0 24.3%

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

      1. *-un-lft-identity 24.3%

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

      2. *-commutative 24.3%

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

   9. Applied egg-rr 30.7%

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

   if 5.4999999999999998e121 < angle

   1. Initial program 28.7%

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

   3. Simplified 32.4%

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

      1. unpow2 32.4%

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

      2. unpow2 32.4%

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

      3. difference-of-squares 32.4%

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

   6. Applied egg-rr 32.4%

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

   7. Taylor expanded in angle around 0 41.5%

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

      1. clear-num 41.5%

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

      2. un-div-inv 41.5%

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

   9. Applied egg-rr 41.5%

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

       1. associate-/r/ 43.1%

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

       2. *-commutative 43.1%

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

   11. Simplified 43.1%

       \[\leadsto 1 \cdot \left(2 \cdot \left(\sin \color{blue}{\left(\pi \cdot \frac{angle}{-180}\right)} \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 16000000000000:\\ \;\;\;\;2 \cdot \left(-0.005555555555555556 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) - angle \cdot \left(\pi \cdot {b}^{2}\right)\right)\right)\\ \mathbf{elif}\;angle \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;2 \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\\ \mathbf{elif}\;angle \leq 5.5 \cdot 10^{+121}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{-180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(a\_m + b\right) \cdot \left(a\_m - b\right)\\ \mathbf{if}\;b \leq 10^{+175}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (+ a_m b) (- a_m b))))
   (if (<= b 1e+175)
     (* 2.0 (* t_0 (sin (* angle (/ PI -180.0)))))
     (* 2.0 (* -0.005555555555555556 (* angle (* PI t_0)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (a_m + b) * (a_m - b);
	double tmp;
	if (b <= 1e+175) {
		tmp = 2.0 * (t_0 * sin((angle * (((double) M_PI) / -180.0))));
	} else {
		tmp = 2.0 * (-0.005555555555555556 * (angle * (((double) M_PI) * t_0)));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = (a_m + b) * (a_m - b);
	double tmp;
	if (b <= 1e+175) {
		tmp = 2.0 * (t_0 * Math.sin((angle * (Math.PI / -180.0))));
	} else {
		tmp = 2.0 * (-0.005555555555555556 * (angle * (Math.PI * t_0)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = (a_m + b) * (a_m - b)
	tmp = 0
	if b <= 1e+175:
		tmp = 2.0 * (t_0 * math.sin((angle * (math.pi / -180.0))))
	else:
		tmp = 2.0 * (-0.005555555555555556 * (angle * (math.pi * t_0)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(a_m + b) * Float64(a_m - b))
	tmp = 0.0
	if (b <= 1e+175)
		tmp = Float64(2.0 * Float64(t_0 * sin(Float64(angle * Float64(pi / -180.0)))));
	else
		tmp = Float64(2.0 * Float64(-0.005555555555555556 * Float64(angle * Float64(pi * t_0))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = (a_m + b) * (a_m - b);
	tmp = 0.0;
	if (b <= 1e+175)
		tmp = 2.0 * (t_0 * sin((angle * (pi / -180.0))));
	else
		tmp = 2.0 * (-0.005555555555555556 * (angle * (pi * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1e+175], N[(2.0 * N[(t$95$0 * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.005555555555555556 * N[(angle * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 9.9999999999999994e174

   1. Initial program 55.2%

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

   3. Simplified 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. difference-of-squares 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in angle around 0 54.3%

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

   if 9.9999999999999994e174 < b

   1. Initial program 36.6%

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

   3. Simplified 42.5%

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

      1. unpow2 42.5%

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

      2. unpow2 42.5%

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

      3. difference-of-squares 51.9%

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

   6. Applied egg-rr 51.9%

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

   7. Taylor expanded in angle around 0 51.9%

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

   8. Taylor expanded in angle around 0 63.7%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+175}:\\ \;\;\;\;2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 2 \cdot \left(\left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (*
  2.0
  (* (* (+ a_m b) (- a_m b)) (sin (* PI (* angle -0.005555555555555556))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return 2.0 * (((a_m + b) * (a_m - b)) * sin((((double) M_PI) * (angle * -0.005555555555555556))));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return 2.0 * (((a_m + b) * (a_m - b)) * Math.sin((Math.PI * (angle * -0.005555555555555556))));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return 2.0 * (((a_m + b) * (a_m - b)) * math.sin((math.pi * (angle * -0.005555555555555556))))

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(2.0 * Float64(Float64(Float64(a_m + b) * Float64(a_m - b)) * sin(Float64(pi * Float64(angle * -0.005555555555555556)))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = 2.0 * (((a_m + b) * (a_m - b)) * sin((pi * (angle * -0.005555555555555556))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(2.0 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8%

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

3. Simplified 54.7%

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

   1. unpow2 54.7%

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

   2. unpow2 54.7%

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

   3. difference-of-squares 57.6%

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in angle around 0 54.0%

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

8. Taylor expanded in angle around inf 55.0%

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

   1. *-commutative 55.0%

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

   2. *-commutative 55.0%

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

   3. associate-*l* 55.4%

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

10. Simplified 55.4%

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

11. Final simplification 55.4%

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

Alternative 14: 56.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(2.0 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8%

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

3. Simplified 54.7%

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

   1. unpow2 54.7%

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

   2. unpow2 54.7%

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

   3. difference-of-squares 57.6%

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in angle around 0 54.0%

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

   1. clear-num 54.0%

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

   2. un-div-inv 53.9%

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

9. Applied egg-rr 53.9%

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

    1. associate-/r/ 56.5%

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

    2. *-commutative 56.5%

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

11. Simplified 56.5%

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

12. Final simplification 56.5%

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

Alternative 15: 54.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 2 \cdot \left(-0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* 2.0 (* -0.005555555555555556 (* angle (* PI (* (+ a_m b) (- a_m b)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return 2.0 * (-0.005555555555555556 * (angle * (((double) M_PI) * ((a_m + b) * (a_m - b)))));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return 2.0 * (-0.005555555555555556 * (angle * (Math.PI * ((a_m + b) * (a_m - b)))));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return 2.0 * (-0.005555555555555556 * (angle * (math.pi * ((a_m + b) * (a_m - b)))))

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(2.0 * Float64(-0.005555555555555556 * Float64(angle * Float64(pi * Float64(Float64(a_m + b) * Float64(a_m - b))))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = 2.0 * (-0.005555555555555556 * (angle * (pi * ((a_m + b) * (a_m - b)))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(2.0 * N[(-0.005555555555555556 * N[(angle * N[(Pi * N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8%

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

3. Simplified 54.7%

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

   1. unpow2 54.7%

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

   2. unpow2 54.7%

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

   3. difference-of-squares 57.6%

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

6. Applied egg-rr 57.6%

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

7. Taylor expanded in angle around 0 54.0%

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

8. Taylor expanded in angle around 0 52.3%

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

9. Final simplification 52.3%

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

Reproduce

herbie shell --seed 2024076 
(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)))))