ab-angle->ABCF B

Percentage Accurate: 54.6% → 68.3%

Time: 31.4s

Alternatives: 13

Speedup: 32.2×

• 32.4% 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: 54.6% 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: 68.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {\left(\sqrt{\pi}\right)}^{2}\\ t_1 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_2 := \sin t\_1\\ t_3 := \left(a\_m + b\right) \cdot \left(a\_m - b\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\cos \left(\frac{angle\_m}{180} \cdot t\_0\right) \cdot \left(2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot t\_2\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+198}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{t\_0}{-180}\right) \cdot \left(2 \cdot \left({\left(\sqrt[3]{t\_2}\right)}^{3} \cdot t\_3\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right)\right) \cdot \sin \left(\frac{angle\_m}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\cos t\_1\right)\right) \cdot \left(2 \cdot \left(t\_3 \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (pow (sqrt PI) 2.0))
        (t_1 (* PI (* angle_m 0.005555555555555556)))
        (t_2 (sin t_1))
        (t_3 (* (+ a_m b) (- a_m b))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+131)
      (*
       (cos (* (/ angle_m 180.0) t_0))
       (* 2.0 (* (+ a_m b) (* (- b a_m) t_2))))
      (if (<= (/ angle_m 180.0) 2e+198)
        (*
         (cos (* angle_m (/ t_0 -180.0)))
         (* 2.0 (* (pow (cbrt t_2) 3.0) t_3)))
        (if (<= (/ angle_m 180.0) 5e+223)
          (*
           (*
            (* 2.0 (* (+ a_m b) (- b a_m)))
            (sin (* (/ angle_m 180.0) (cbrt (pow PI 3.0)))))
           (cos (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0)))))
          (*
           (expm1 (log1p (cos t_1)))
           (* 2.0 (* t_3 (sin (* angle_m (/ PI -180.0))))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = pow(sqrt(((double) M_PI)), 2.0);
	double t_1 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_2 = sin(t_1);
	double t_3 = (a_m + b) * (a_m - b);
	double tmp;
	if ((angle_m / 180.0) <= 5e+131) {
		tmp = cos(((angle_m / 180.0) * t_0)) * (2.0 * ((a_m + b) * ((b - a_m) * t_2)));
	} else if ((angle_m / 180.0) <= 2e+198) {
		tmp = cos((angle_m * (t_0 / -180.0))) * (2.0 * (pow(cbrt(t_2), 3.0) * t_3));
	} else if ((angle_m / 180.0) <= 5e+223) {
		tmp = ((2.0 * ((a_m + b) * (b - a_m))) * sin(((angle_m / 180.0) * cbrt(pow(((double) M_PI), 3.0))))) * cos(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))));
	} else {
		tmp = expm1(log1p(cos(t_1))) * (2.0 * (t_3 * sin((angle_m * (((double) M_PI) / -180.0)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.pow(Math.sqrt(Math.PI), 2.0);
	double t_1 = Math.PI * (angle_m * 0.005555555555555556);
	double t_2 = Math.sin(t_1);
	double t_3 = (a_m + b) * (a_m - b);
	double tmp;
	if ((angle_m / 180.0) <= 5e+131) {
		tmp = Math.cos(((angle_m / 180.0) * t_0)) * (2.0 * ((a_m + b) * ((b - a_m) * t_2)));
	} else if ((angle_m / 180.0) <= 2e+198) {
		tmp = Math.cos((angle_m * (t_0 / -180.0))) * (2.0 * (Math.pow(Math.cbrt(t_2), 3.0) * t_3));
	} else if ((angle_m / 180.0) <= 5e+223) {
		tmp = ((2.0 * ((a_m + b) * (b - a_m))) * Math.sin(((angle_m / 180.0) * Math.cbrt(Math.pow(Math.PI, 3.0))))) * Math.cos(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))));
	} else {
		tmp = Math.expm1(Math.log1p(Math.cos(t_1))) * (2.0 * (t_3 * Math.sin((angle_m * (Math.PI / -180.0)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = sqrt(pi) ^ 2.0
	t_1 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_2 = sin(t_1)
	t_3 = Float64(Float64(a_m + b) * Float64(a_m - b))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+131)
		tmp = Float64(cos(Float64(Float64(angle_m / 180.0) * t_0)) * Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * t_2))));
	elseif (Float64(angle_m / 180.0) <= 2e+198)
		tmp = Float64(cos(Float64(angle_m * Float64(t_0 / -180.0))) * Float64(2.0 * Float64((cbrt(t_2) ^ 3.0) * t_3)));
	elseif (Float64(angle_m / 180.0) <= 5e+223)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(a_m + b) * Float64(b - a_m))) * sin(Float64(Float64(angle_m / 180.0) * cbrt((pi ^ 3.0))))) * cos(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))));
	else
		tmp = Float64(expm1(log1p(cos(t_1))) * Float64(2.0 * Float64(t_3 * sin(Float64(angle_m * Float64(pi / -180.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+131], N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+198], N[(N[Cos[N[(angle$95$m * N[(t$95$0 / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+223], N[(N[(N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[Log[1 + N[Cos[t$95$1], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(2.0 * N[(t$95$3 * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999995e131

   1. Initial program 62.5%

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

      1. unpow2 62.5%

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

      2. unpow2 62.5%

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

      3. difference-of-squares 65.5%

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in angle around inf 62.8%

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

      1. *-commutative 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-*r* 63.0%

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

      4. *-commutative 63.0%

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

      5. associate-*l* 74.0%

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

   7. Simplified 74.0%

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

      1. add-sqr-sqrt 75.6%

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

      2. pow2 75.6%

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

   9. Applied egg-rr 75.6%

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

   if 4.99999999999999995e131 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000004e198

   1. Initial program 24.5%

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

   3. Simplified 29.0%

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

      1. unpow2 29.0%

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

      2. unpow2 29.0%

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

      3. difference-of-squares 29.0%

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

   6. Applied egg-rr 29.0%

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

      1. add-cube-cbrt 29.0%

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

      2. pow3 29.0%

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

   8. Applied egg-rr 36.5%

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

      1. add-sqr-sqrt 23.9%

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

      2. pow2 23.9%

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

   10. Applied egg-rr 43.5%

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

   if 2.00000000000000004e198 < (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999985e223

   1. Initial program 33.5%

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

      1. unpow2 33.5%

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

      2. unpow2 33.5%

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

      3. difference-of-squares 33.5%

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

   4. Applied egg-rr 33.5%

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

      1. add-cube-cbrt 31.3%

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

      2. pow2 31.3%

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

   6. Applied egg-rr 31.3%

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

      1. add-cbrt-cube 73.4%

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

      2. pow3 73.4%

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

   8. Applied egg-rr 73.4%

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

   if 4.99999999999999985e223 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 30.2%

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

   3. Simplified 34.7%

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

      1. unpow2 34.7%

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

      2. unpow2 34.7%

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

      3. difference-of-squares 40.6%

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

   6. Applied egg-rr 40.6%

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

      1. expm1-log1p-u 40.6%

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

      2. expm1-undefine 40.6%

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

   8. Applied egg-rr 52.4%

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

      1. expm1-define 52.4%

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

   10. Simplified 52.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \left(2 \cdot \left(\sin \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right) \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+198}:\\ \;\;\;\;\cos \left(angle \cdot \frac{{\left(\sqrt{\pi}\right)}^{2}}{-180}\right) \cdot \left(2 \cdot \left({\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \sqrt[3]{{\pi}^{3}}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+118}:\\ \;\;\;\;\cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right) \cdot \left(2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right) \cdot \sin \left(angle\_m \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(angle\_m \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+118)
    (*
     (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0)))
     (*
      2.0
      (*
       (+ a_m b)
       (* (- b a_m) (sin (* PI (* angle_m 0.005555555555555556)))))))
    (*
     (* 2.0 (* (* (+ a_m b) (- a_m b)) (sin (* angle_m (/ PI -180.0)))))
     (cos (* angle_m (/ (* (cbrt PI) (pow (cbrt PI) 2.0)) -180.0)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+118) {
		tmp = cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0))) * (2.0 * ((a_m + b) * ((b - a_m) * sin((((double) M_PI) * (angle_m * 0.005555555555555556))))));
	} else {
		tmp = (2.0 * (((a_m + b) * (a_m - b)) * sin((angle_m * (((double) M_PI) / -180.0))))) * cos((angle_m * ((cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)) / -180.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+118) {
		tmp = Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0))) * (2.0 * ((a_m + b) * ((b - a_m) * Math.sin((Math.PI * (angle_m * 0.005555555555555556))))));
	} else {
		tmp = (2.0 * (((a_m + b) * (a_m - b)) * Math.sin((angle_m * (Math.PI / -180.0))))) * Math.cos((angle_m * ((Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)) / -180.0)));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+118)
		tmp = Float64(cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0))) * Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(a_m + b) * Float64(a_m - b)) * sin(Float64(angle_m * Float64(pi / -180.0))))) * cos(Float64(angle_m * Float64(Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)) / -180.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+118], N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle$95$m * N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999967e117

   1. Initial program 62.3%

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

      1. unpow2 62.3%

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

      2. unpow2 62.3%

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

      3. difference-of-squares 65.3%

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

   4. Applied egg-rr 65.3%

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

   5. Taylor expanded in angle around inf 62.5%

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

      1. *-commutative 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-*r* 62.8%

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

      4. *-commutative 62.8%

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

      5. associate-*l* 74.0%

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

   7. Simplified 74.0%

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

      1. add-sqr-sqrt 75.5%

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

      2. pow2 75.5%

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

   9. Applied egg-rr 75.5%

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

   if 9.99999999999999967e117 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.8%

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

   3. Simplified 35.5%

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

      1. unpow2 35.5%

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

      2. unpow2 35.5%

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

      3. difference-of-squares 37.4%

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

   6. Applied egg-rr 37.4%

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

      1. add-cube-cbrt 39.1%

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

      2. pow2 39.1%

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

   8. Applied egg-rr 47.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+118}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right) \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(angle \cdot \frac{\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}}{-180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := 2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin t\_0\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a\_m}^{2} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\cos t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* 2.0 (* (+ a_m b) (* (- b a_m) (sin t_0))))))
   (*
    angle_s
    (if (<= (pow a_m 2.0) 5e+94)
      (* (cos t_0) t_1)
      (*
       t_1
       (cos (* (* angle_m 0.005555555555555556) (cbrt (pow PI 3.0)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * ((a_m + b) * ((b - a_m) * sin(t_0)));
	double tmp;
	if (pow(a_m, 2.0) <= 5e+94) {
		tmp = cos(t_0) * t_1;
	} else {
		tmp = t_1 * cos(((angle_m * 0.005555555555555556) * cbrt(pow(((double) M_PI), 3.0))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * ((a_m + b) * ((b - a_m) * Math.sin(t_0)));
	double tmp;
	if (Math.pow(a_m, 2.0) <= 5e+94) {
		tmp = Math.cos(t_0) * t_1;
	} else {
		tmp = t_1 * Math.cos(((angle_m * 0.005555555555555556) * Math.cbrt(Math.pow(Math.PI, 3.0))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(t_0))))
	tmp = 0.0
	if ((a_m ^ 2.0) <= 5e+94)
		tmp = Float64(cos(t_0) * t_1);
	else
		tmp = Float64(t_1 * cos(Float64(Float64(angle_m * 0.005555555555555556) * cbrt((pi ^ 3.0)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[a$95$m, 2.0], $MachinePrecision], 5e+94], N[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[Cos[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 5.0000000000000001e94

   1. Initial program 59.0%

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

      1. unpow2 59.0%

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

      2. unpow2 59.0%

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

      3. difference-of-squares 59.0%

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

   4. Applied egg-rr 59.0%

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

   5. Taylor expanded in angle around inf 56.3%

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

      1. *-commutative 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*r* 56.4%

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

      4. *-commutative 56.4%

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

      5. associate-*l* 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in angle around inf 57.8%

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

      1. *-commutative 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*r* 62.3%

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

   10. Simplified 62.3%

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

   if 5.0000000000000001e94 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 52.2%

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

      1. unpow2 52.2%

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

      2. unpow2 52.2%

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

      3. difference-of-squares 59.0%

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

   4. Applied egg-rr 59.0%

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

   5. Taylor expanded in angle around inf 57.8%

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

      1. *-commutative 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*r* 58.8%

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

      4. *-commutative 58.8%

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

      5. associate-*l* 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in angle around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-*r* 74.9%

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

   10. Simplified 74.9%

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

       1. add-cbrt-cube 58.9%

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

       2. pow3 58.9%

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

   12. Applied egg-rr 79.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{+161}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a\_m \cdot \left(a\_m \cdot t\_0\right) - t\_0 \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+263}:\\ \;\;\;\;2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* -0.005555555555555556 (* angle_m PI)))))
   (*
    angle_s
    (if (<= b 4.2e+161)
      (*
       (cos (* angle_m (/ PI -180.0)))
       (* 2.0 (- (* a_m (* a_m t_0)) (* t_0 (pow b 2.0)))))
      (if (<= b 1.3e+263)
        (*
         2.0
         (*
          (+ a_m b)
          (* (- b a_m) (sin (* PI (* angle_m 0.005555555555555556))))))
        (*
         0.011111111111111112
         (* angle_m (* PI (* (+ a_m b) (- b a_m))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = sin((-0.005555555555555556 * (angle_m * ((double) M_PI))));
	double tmp;
	if (b <= 4.2e+161) {
		tmp = cos((angle_m * (((double) M_PI) / -180.0))) * (2.0 * ((a_m * (a_m * t_0)) - (t_0 * pow(b, 2.0))));
	} else if (b <= 1.3e+263) {
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((a_m + b) * (b - a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.sin((-0.005555555555555556 * (angle_m * Math.PI)));
	double tmp;
	if (b <= 4.2e+161) {
		tmp = Math.cos((angle_m * (Math.PI / -180.0))) * (2.0 * ((a_m * (a_m * t_0)) - (t_0 * Math.pow(b, 2.0))));
	} else if (b <= 1.3e+263) {
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * Math.sin((Math.PI * (angle_m * 0.005555555555555556)))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((a_m + b) * (b - a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = math.sin((-0.005555555555555556 * (angle_m * math.pi)))
	tmp = 0
	if b <= 4.2e+161:
		tmp = math.cos((angle_m * (math.pi / -180.0))) * (2.0 * ((a_m * (a_m * t_0)) - (t_0 * math.pow(b, 2.0))))
	elif b <= 1.3e+263:
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * math.sin((math.pi * (angle_m * 0.005555555555555556)))))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((a_m + b) * (b - a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = sin(Float64(-0.005555555555555556 * Float64(angle_m * pi)))
	tmp = 0.0
	if (b <= 4.2e+161)
		tmp = Float64(cos(Float64(angle_m * Float64(pi / -180.0))) * Float64(2.0 * Float64(Float64(a_m * Float64(a_m * t_0)) - Float64(t_0 * (b ^ 2.0)))));
	elseif (b <= 1.3e+263)
		tmp = Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a_m + b) * Float64(b - a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = sin((-0.005555555555555556 * (angle_m * pi)));
	tmp = 0.0;
	if (b <= 4.2e+161)
		tmp = cos((angle_m * (pi / -180.0))) * (2.0 * ((a_m * (a_m * t_0)) - (t_0 * (b ^ 2.0))));
	elseif (b <= 1.3e+263)
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * sin((pi * (angle_m * 0.005555555555555556)))));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * ((a_m + b) * (b - a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 4.2e+161], N[(N[Cos[N[(angle$95$m * 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, 1.3e+263], N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if b < 4.2e161

   1. Initial program 59.1%

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

   3. Simplified 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. difference-of-squares 60.6%

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

   6. Applied egg-rr 60.6%

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

   7. Taylor expanded in a around 0 66.2%

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

      1. +-commutative 66.2%

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

      2. mul-1-neg 66.2%

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

      3. unsub-neg 66.2%

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

      4. *-commutative 66.2%

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

   9. Simplified 66.2%

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

   if 4.2e161 < b < 1.3000000000000001e263

   1. Initial program 21.1%

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

      1. unpow2 21.1%

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

      2. unpow2 21.1%

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

      3. difference-of-squares 36.4%

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

   4. Applied egg-rr 36.4%

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

   5. Taylor expanded in angle around inf 26.4%

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

      1. *-commutative 26.4%

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

      2. *-commutative 26.4%

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

      3. associate-*r* 36.4%

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

      4. *-commutative 36.4%

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

      5. associate-*l* 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in angle around inf 49.8%

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

      1. *-commutative 49.8%

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

      2. *-commutative 49.8%

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

      3. associate-*r* 54.8%

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

   10. Simplified 54.8%

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

   11. Taylor expanded in angle around 0 89.8%

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

   if 1.3000000000000001e263 < b

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*l* 60.8%

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

   3. Simplified 60.8%

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

      1. unpow2 60.8%

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

      2. unpow2 60.8%

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

      3. difference-of-squares 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0 70.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{+161}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(a \cdot \left(a \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) - \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+263}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a\_m + b\right) \cdot \left(a\_m - b\right)\\ t_1 := angle\_m \cdot \frac{\pi}{-180}\\ t_2 := \cos t\_1\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(\left(a\_m + b\right) \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(\left(b - a\_m\right) \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+103} \lor \neg \left(\frac{angle\_m}{180} \leq 10^{+138}\right):\\ \;\;\;\;\left(2 \cdot \left(t\_0 \cdot \sin t\_1\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(2 \cdot \left(-0.005555555555555556 \cdot \left(t\_0 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a_m b) (- a_m b)))
        (t_1 (* angle_m (/ PI -180.0)))
        (t_2 (cos t_1)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e-26)
      (*
       (cos (* (/ angle_m 180.0) PI))
       (*
        2.0
        (* (+ a_m b) (* 0.005555555555555556 (* angle_m (* (- b a_m) PI))))))
      (if (or (<= (/ angle_m 180.0) 1e+103)
              (not (<= (/ angle_m 180.0) 1e+138)))
        (* (* 2.0 (* t_0 (sin t_1))) t_2)
        (* t_2 (* 2.0 (* -0.005555555555555556 (* t_0 (* angle_m PI))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (a_m + b) * (a_m - b);
	double t_1 = angle_m * (((double) M_PI) / -180.0);
	double t_2 = cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 5e-26) {
		tmp = cos(((angle_m / 180.0) * ((double) M_PI))) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * ((double) M_PI))))));
	} else if (((angle_m / 180.0) <= 1e+103) || !((angle_m / 180.0) <= 1e+138)) {
		tmp = (2.0 * (t_0 * sin(t_1))) * t_2;
	} else {
		tmp = t_2 * (2.0 * (-0.005555555555555556 * (t_0 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (a_m + b) * (a_m - b);
	double t_1 = angle_m * (Math.PI / -180.0);
	double t_2 = Math.cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 5e-26) {
		tmp = Math.cos(((angle_m / 180.0) * Math.PI)) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * Math.PI)))));
	} else if (((angle_m / 180.0) <= 1e+103) || !((angle_m / 180.0) <= 1e+138)) {
		tmp = (2.0 * (t_0 * Math.sin(t_1))) * t_2;
	} else {
		tmp = t_2 * (2.0 * (-0.005555555555555556 * (t_0 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = (a_m + b) * (a_m - b)
	t_1 = angle_m * (math.pi / -180.0)
	t_2 = math.cos(t_1)
	tmp = 0
	if (angle_m / 180.0) <= 5e-26:
		tmp = math.cos(((angle_m / 180.0) * math.pi)) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * math.pi)))))
	elif ((angle_m / 180.0) <= 1e+103) or not ((angle_m / 180.0) <= 1e+138):
		tmp = (2.0 * (t_0 * math.sin(t_1))) * t_2
	else:
		tmp = t_2 * (2.0 * (-0.005555555555555556 * (t_0 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(a_m + b) * Float64(a_m - b))
	t_1 = Float64(angle_m * Float64(pi / -180.0))
	t_2 = cos(t_1)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-26)
		tmp = Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(2.0 * Float64(Float64(a_m + b) * Float64(0.005555555555555556 * Float64(angle_m * Float64(Float64(b - a_m) * pi))))));
	elseif ((Float64(angle_m / 180.0) <= 1e+103) || !(Float64(angle_m / 180.0) <= 1e+138))
		tmp = Float64(Float64(2.0 * Float64(t_0 * sin(t_1))) * t_2);
	else
		tmp = Float64(t_2 * Float64(2.0 * Float64(-0.005555555555555556 * Float64(t_0 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = (a_m + b) * (a_m - b);
	t_1 = angle_m * (pi / -180.0);
	t_2 = cos(t_1);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-26)
		tmp = cos(((angle_m / 180.0) * pi)) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * pi)))));
	elseif (((angle_m / 180.0) <= 1e+103) || ~(((angle_m / 180.0) <= 1e+138)))
		tmp = (2.0 * (t_0 * sin(t_1))) * t_2;
	else
		tmp = t_2 * (2.0 * (-0.005555555555555556 * (t_0 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-26], N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(0.005555555555555556 * N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+103], N[Not[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+138]], $MachinePrecision]], N[(N[(2.0 * N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$2 * N[(2.0 * N[(-0.005555555555555556 * N[(t$95$0 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000019e-26

   1. Initial program 63.9%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. difference-of-squares 66.2%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in angle around inf 65.1%

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

      1. *-commutative 65.1%

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

      2. *-commutative 65.1%

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

      3. associate-*r* 64.5%

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

      4. *-commutative 64.5%

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

      5. associate-*l* 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in angle around 0 73.1%

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

   if 5.00000000000000019e-26 < (/.f64 angle #s(literal 180 binary64)) < 1e103 or 1e138 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 38.3%

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

   3. Simplified 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 46.8%

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

   6. Applied egg-rr 46.8%

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

   if 1e103 < (/.f64 angle #s(literal 180 binary64)) < 1e138

   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 43.2%

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

      1. unpow2 43.2%

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

      2. unpow2 43.2%

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

      3. difference-of-squares 43.2%

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

   6. Applied egg-rr 43.2%

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

   7. Taylor expanded in angle around 0 86.4%

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

      1. associate-*r* 86.4%

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

   9. Simplified 86.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+103} \lor \neg \left(\frac{angle}{180} \leq 10^{+138}\right):\\ \;\;\;\;\left(2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)\right) \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(-0.005555555555555556 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := 2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin t\_0\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+164}:\\ \;\;\;\;\cos t\_0 \cdot t\_1\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+262}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* 2.0 (* (+ a_m b) (* (- b a_m) (sin t_0))))))
   (*
    angle_s
    (if (<= b 1.35e+164)
      (* (cos t_0) t_1)
      (if (<= b 6.6e+262)
        t_1
        (*
         0.011111111111111112
         (* angle_m (* PI (* (+ a_m b) (- b a_m))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * ((a_m + b) * ((b - a_m) * sin(t_0)));
	double tmp;
	if (b <= 1.35e+164) {
		tmp = cos(t_0) * t_1;
	} else if (b <= 6.6e+262) {
		tmp = t_1;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((a_m + b) * (b - a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * ((a_m + b) * ((b - a_m) * Math.sin(t_0)));
	double tmp;
	if (b <= 1.35e+164) {
		tmp = Math.cos(t_0) * t_1;
	} else if (b <= 6.6e+262) {
		tmp = t_1;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((a_m + b) * (b - a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	t_1 = 2.0 * ((a_m + b) * ((b - a_m) * math.sin(t_0)))
	tmp = 0
	if b <= 1.35e+164:
		tmp = math.cos(t_0) * t_1
	elif b <= 6.6e+262:
		tmp = t_1
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((a_m + b) * (b - a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(t_0))))
	tmp = 0.0
	if (b <= 1.35e+164)
		tmp = Float64(cos(t_0) * t_1);
	elseif (b <= 6.6e+262)
		tmp = t_1;
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a_m + b) * Float64(b - a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = pi * (angle_m * 0.005555555555555556);
	t_1 = 2.0 * ((a_m + b) * ((b - a_m) * sin(t_0)));
	tmp = 0.0;
	if (b <= 1.35e+164)
		tmp = cos(t_0) * t_1;
	elseif (b <= 6.6e+262)
		tmp = t_1;
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * ((a_m + b) * (b - a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 1.35e+164], N[(N[Cos[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[b, 6.6e+262], t$95$1, N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.35000000000000003e164

   1. Initial program 59.1%

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

      1. unpow2 59.1%

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

      2. unpow2 59.1%

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

      3. difference-of-squares 60.5%

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

   4. Applied egg-rr 60.5%

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

   5. Taylor expanded in angle around inf 59.4%

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

      1. *-commutative 59.4%

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

      2. *-commutative 59.4%

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

      3. associate-*r* 59.1%

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

      4. *-commutative 59.1%

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

      5. associate-*l* 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in angle around inf 67.8%

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

      1. *-commutative 67.8%

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

      2. *-commutative 67.8%

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

      3. associate-*r* 68.0%

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

   10. Simplified 68.0%

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

   if 1.35000000000000003e164 < b < 6.59999999999999997e262

   1. Initial program 21.1%

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

      1. unpow2 21.1%

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

      2. unpow2 21.1%

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

      3. difference-of-squares 36.4%

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

   4. Applied egg-rr 36.4%

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

   5. Taylor expanded in angle around inf 26.4%

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

      1. *-commutative 26.4%

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

      2. *-commutative 26.4%

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

      3. associate-*r* 36.4%

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

      4. *-commutative 36.4%

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

      5. associate-*l* 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in angle around inf 49.8%

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

      1. *-commutative 49.8%

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

      2. *-commutative 49.8%

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

      3. associate-*r* 54.8%

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

   10. Simplified 54.8%

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

   11. Taylor expanded in angle around 0 89.8%

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

   if 6.59999999999999997e262 < b

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*l* 60.8%

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

   3. Simplified 60.8%

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

      1. unpow2 60.8%

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

      2. unpow2 60.8%

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

      3. difference-of-squares 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0 70.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+164}:\\ \;\;\;\;\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+262}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+162}:\\ \;\;\;\;t\_0 \cdot \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+262}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0
         (*
          2.0
          (*
           (+ a_m b)
           (* (- b a_m) (sin (* PI (* angle_m 0.005555555555555556))))))))
   (*
    angle_s
    (if (<= b 5e+162)
      (* t_0 (cos (* (/ angle_m 180.0) PI)))
      (if (<= b 5e+262)
        t_0
        (*
         0.011111111111111112
         (* angle_m (* PI (* (+ a_m b) (- b a_m))))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((a_m + b) * ((b - a_m) * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))));
	double tmp;
	if (b <= 5e+162) {
		tmp = t_0 * cos(((angle_m / 180.0) * ((double) M_PI)));
	} else if (b <= 5e+262) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((a_m + b) * (b - a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * ((a_m + b) * ((b - a_m) * Math.sin((Math.PI * (angle_m * 0.005555555555555556)))));
	double tmp;
	if (b <= 5e+162) {
		tmp = t_0 * Math.cos(((angle_m / 180.0) * Math.PI));
	} else if (b <= 5e+262) {
		tmp = t_0;
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((a_m + b) * (b - a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * ((a_m + b) * ((b - a_m) * math.sin((math.pi * (angle_m * 0.005555555555555556)))))
	tmp = 0
	if b <= 5e+162:
		tmp = t_0 * math.cos(((angle_m / 180.0) * math.pi))
	elif b <= 5e+262:
		tmp = t_0
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((a_m + b) * (b - a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))))
	tmp = 0.0
	if (b <= 5e+162)
		tmp = Float64(t_0 * cos(Float64(Float64(angle_m / 180.0) * pi)));
	elseif (b <= 5e+262)
		tmp = t_0;
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a_m + b) * Float64(b - a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((a_m + b) * ((b - a_m) * sin((pi * (angle_m * 0.005555555555555556)))));
	tmp = 0.0;
	if (b <= 5e+162)
		tmp = t_0 * cos(((angle_m / 180.0) * pi));
	elseif (b <= 5e+262)
		tmp = t_0;
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * ((a_m + b) * (b - a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 5e+162], N[(t$95$0 * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+262], t$95$0, N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if b < 4.9999999999999997e162

   1. Initial program 59.1%

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

      1. unpow2 59.1%

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

      2. unpow2 59.1%

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

      3. difference-of-squares 60.5%

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

   4. Applied egg-rr 60.5%

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

   5. Taylor expanded in angle around inf 59.4%

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

      1. *-commutative 59.4%

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

      2. *-commutative 59.4%

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

      3. associate-*r* 59.1%

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

      4. *-commutative 59.1%

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

      5. associate-*l* 67.3%

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

   7. Simplified 67.3%

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

   if 4.9999999999999997e162 < b < 5.00000000000000008e262

   1. Initial program 21.1%

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

      1. unpow2 21.1%

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

      2. unpow2 21.1%

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

      3. difference-of-squares 36.4%

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

   4. Applied egg-rr 36.4%

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

   5. Taylor expanded in angle around inf 26.4%

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

      1. *-commutative 26.4%

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

      2. *-commutative 26.4%

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

      3. associate-*r* 36.4%

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

      4. *-commutative 36.4%

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

      5. associate-*l* 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in angle around inf 49.8%

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

      1. *-commutative 49.8%

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

      2. *-commutative 49.8%

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

      3. associate-*r* 54.8%

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

   10. Simplified 54.8%

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

   11. Taylor expanded in angle around 0 89.8%

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

   if 5.00000000000000008e262 < b

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*l* 60.8%

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

   3. Simplified 60.8%

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

      1. unpow2 60.8%

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

      2. unpow2 60.8%

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

      3. difference-of-squares 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0 70.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\left(2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+262}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(-0.005555555555555556 \cdot \left(\left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right) \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e-25)
    (*
     2.0
     (* (+ a_m b) (* (- b a_m) (sin (* PI (* angle_m 0.005555555555555556))))))
    (if (<= (/ angle_m 180.0) 2e+45)
      (* (* (+ a_m b) (- b a_m)) (sin (* (* angle_m PI) 0.011111111111111112)))
      (*
       (cos (* angle_m (/ PI -180.0)))
       (*
        2.0
        (*
         -0.005555555555555556
         (* (* (+ a_m b) (- a_m b)) (* angle_m PI)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 2e+45) {
		tmp = ((a_m + b) * (b - a_m)) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = cos((angle_m * (((double) M_PI) / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-25) {
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * Math.sin((Math.PI * (angle_m * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 2e+45) {
		tmp = ((a_m + b) * (b - a_m)) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112));
	} else {
		tmp = Math.cos((angle_m * (Math.PI / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-25:
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * math.sin((math.pi * (angle_m * 0.005555555555555556)))))
	elif (angle_m / 180.0) <= 2e+45:
		tmp = ((a_m + b) * (b - a_m)) * math.sin(((angle_m * math.pi) * 0.011111111111111112))
	else:
		tmp = math.cos((angle_m * (math.pi / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * math.pi))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-25)
		tmp = Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))));
	elseif (Float64(angle_m / 180.0) <= 2e+45)
		tmp = Float64(Float64(Float64(a_m + b) * Float64(b - a_m)) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)));
	else
		tmp = Float64(cos(Float64(angle_m * Float64(pi / -180.0))) * Float64(2.0 * Float64(-0.005555555555555556 * Float64(Float64(Float64(a_m + b) * Float64(a_m - b)) * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-25)
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * sin((pi * (angle_m * 0.005555555555555556)))));
	elseif ((angle_m / 180.0) <= 2e+45)
		tmp = ((a_m + b) * (b - a_m)) * sin(((angle_m * pi) * 0.011111111111111112));
	else
		tmp = cos((angle_m * (pi / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-25], N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+45], N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(-0.005555555555555556 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999962e-25

   1. Initial program 63.9%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. difference-of-squares 66.2%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in angle around inf 65.1%

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

      1. *-commutative 65.1%

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

      2. *-commutative 65.1%

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

      3. associate-*r* 64.5%

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

      4. *-commutative 64.5%

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

      5. associate-*l* 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in angle around inf 75.6%

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

      1. *-commutative 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-*r* 77.8%

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

   10. Simplified 77.8%

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

   11. Taylor expanded in angle around 0 75.4%

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

   if 4.99999999999999962e-25 < (/.f64 angle #s(literal 180 binary64)) < 1.9999999999999999e45

   1. Initial program 54.4%

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

      1. associate-*l* 54.4%

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

      2. *-commutative 54.4%

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

      3. associate-*l* 54.4%

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

   3. Simplified 54.4%

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

      1. *-commutative 54.4%

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

      2. sub-neg 54.4%

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

      3. distribute-lft-in 54.4%

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

      4. 2-sin 54.4%

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

      5. div-inv 54.4%

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

      6. metadata-eval 54.4%

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

   6. Applied egg-rr 54.4%

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

      1. distribute-lft-out 54.4%

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

      2. sub-neg 54.4%

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

      3. *-commutative 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-*r* 61.5%

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

      6. *-commutative 61.5%

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

      7. associate-*l* 61.5%

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

      8. metadata-eval 61.5%

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

   8. Simplified 61.5%

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

      1. unpow2 54.4%

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

      2. unpow2 54.4%

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

      3. difference-of-squares 66.9%

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

   10. Applied egg-rr 74.0%

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

   if 1.9999999999999999e45 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 34.7%

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

   3. Simplified 37.9%

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

      1. unpow2 37.9%

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

      2. unpow2 37.9%

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

      3. difference-of-squares 39.6%

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

   6. Applied egg-rr 39.6%

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

   7. Taylor expanded in angle around 0 45.1%

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

      1. associate-*r* 45.1%

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

   9. Simplified 45.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(-0.005555555555555556 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(\left(a\_m + b\right) \cdot \left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(\left(b - a\_m\right) \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle\_m \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(-0.005555555555555556 \cdot \left(\left(\left(a\_m + b\right) \cdot \left(a\_m - b\right)\right) \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e-19)
    (*
     (cos (* (/ angle_m 180.0) PI))
     (*
      2.0
      (* (+ a_m b) (* 0.005555555555555556 (* angle_m (* (- b a_m) PI))))))
    (if (<= (/ angle_m 180.0) 2e+45)
      (* (* (+ a_m b) (- b a_m)) (sin (* (* angle_m PI) 0.011111111111111112)))
      (*
       (cos (* angle_m (/ PI -180.0)))
       (*
        2.0
        (*
         -0.005555555555555556
         (* (* (+ a_m b) (- a_m b)) (* angle_m PI)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-19) {
		tmp = cos(((angle_m / 180.0) * ((double) M_PI))) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * ((double) M_PI))))));
	} else if ((angle_m / 180.0) <= 2e+45) {
		tmp = ((a_m + b) * (b - a_m)) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = cos((angle_m * (((double) M_PI) / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-19) {
		tmp = Math.cos(((angle_m / 180.0) * Math.PI)) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * Math.PI)))));
	} else if ((angle_m / 180.0) <= 2e+45) {
		tmp = ((a_m + b) * (b - a_m)) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112));
	} else {
		tmp = Math.cos((angle_m * (Math.PI / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-19:
		tmp = math.cos(((angle_m / 180.0) * math.pi)) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * math.pi)))))
	elif (angle_m / 180.0) <= 2e+45:
		tmp = ((a_m + b) * (b - a_m)) * math.sin(((angle_m * math.pi) * 0.011111111111111112))
	else:
		tmp = math.cos((angle_m * (math.pi / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * math.pi))))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-19)
		tmp = Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(2.0 * Float64(Float64(a_m + b) * Float64(0.005555555555555556 * Float64(angle_m * Float64(Float64(b - a_m) * pi))))));
	elseif (Float64(angle_m / 180.0) <= 2e+45)
		tmp = Float64(Float64(Float64(a_m + b) * Float64(b - a_m)) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)));
	else
		tmp = Float64(cos(Float64(angle_m * Float64(pi / -180.0))) * Float64(2.0 * Float64(-0.005555555555555556 * Float64(Float64(Float64(a_m + b) * Float64(a_m - b)) * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-19)
		tmp = cos(((angle_m / 180.0) * pi)) * (2.0 * ((a_m + b) * (0.005555555555555556 * (angle_m * ((b - a_m) * pi)))));
	elseif ((angle_m / 180.0) <= 2e+45)
		tmp = ((a_m + b) * (b - a_m)) * sin(((angle_m * pi) * 0.011111111111111112));
	else
		tmp = cos((angle_m * (pi / -180.0))) * (2.0 * (-0.005555555555555556 * (((a_m + b) * (a_m - b)) * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-19], N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(0.005555555555555556 * N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+45], N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(-0.005555555555555556 * N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000004e-19

   1. Initial program 63.7%

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

      1. unpow2 63.7%

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

      2. unpow2 63.7%

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

      3. difference-of-squares 66.6%

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

   4. Applied egg-rr 66.6%

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

   5. Taylor expanded in angle around inf 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 64.8%

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

      4. *-commutative 64.8%

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

      5. associate-*l* 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in angle around 0 73.4%

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

   if 5.0000000000000004e-19 < (/.f64 angle #s(literal 180 binary64)) < 1.9999999999999999e45

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. *-commutative 55.1%

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

      3. associate-*l* 55.1%

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

   3. Simplified 55.1%

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

      1. *-commutative 55.1%

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

      2. sub-neg 55.1%

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

      3. distribute-lft-in 55.1%

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

      4. 2-sin 55.1%

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

      5. div-inv 55.1%

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

      6. metadata-eval 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. distribute-lft-out 55.1%

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

      2. sub-neg 55.1%

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

      3. *-commutative 55.1%

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

      4. *-commutative 55.1%

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

      5. associate-*r* 63.3%

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

      6. *-commutative 63.3%

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

      7. associate-*l* 63.3%

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

      8. metadata-eval 63.3%

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

   8. Simplified 63.3%

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

      1. unpow2 55.1%

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

      2. unpow2 55.1%

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

      3. difference-of-squares 62.3%

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

   10. Applied egg-rr 70.4%

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

   if 1.9999999999999999e45 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 34.7%

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

   3. Simplified 37.9%

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

      1. unpow2 37.9%

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

      2. unpow2 37.9%

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

      3. difference-of-squares 39.6%

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

   6. Applied egg-rr 39.6%

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

   7. Taylor expanded in angle around 0 45.1%

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

      1. associate-*r* 45.1%

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

   9. Simplified 45.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(2 \cdot \left(-0.005555555555555556 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-25} \lor \neg \left(\frac{angle\_m}{180} \leq 5 \cdot 10^{+56}\right):\\ \;\;\;\;2 \cdot \left(\left(a\_m + b\right) \cdot \left(\left(b - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (or (<= (/ angle_m 180.0) 5e-25) (not (<= (/ angle_m 180.0) 5e+56)))
    (*
     2.0
     (* (+ a_m b) (* (- b a_m) (sin (* PI (* angle_m 0.005555555555555556))))))
    (*
     (* (+ a_m b) (- b a_m))
     (sin (* (* angle_m PI) 0.011111111111111112))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (((angle_m / 180.0) <= 5e-25) || !((angle_m / 180.0) <= 5e+56)) {
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))));
	} else {
		tmp = ((a_m + b) * (b - a_m)) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (((angle_m / 180.0) <= 5e-25) || !((angle_m / 180.0) <= 5e+56)) {
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * Math.sin((Math.PI * (angle_m * 0.005555555555555556)))));
	} else {
		tmp = ((a_m + b) * (b - a_m)) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if ((angle_m / 180.0) <= 5e-25) or not ((angle_m / 180.0) <= 5e+56):
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * math.sin((math.pi * (angle_m * 0.005555555555555556)))))
	else:
		tmp = ((a_m + b) * (b - a_m)) * math.sin(((angle_m * math.pi) * 0.011111111111111112))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if ((Float64(angle_m / 180.0) <= 5e-25) || !(Float64(angle_m / 180.0) <= 5e+56))
		tmp = Float64(2.0 * Float64(Float64(a_m + b) * Float64(Float64(b - a_m) * sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))));
	else
		tmp = Float64(Float64(Float64(a_m + b) * Float64(b - a_m)) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (((angle_m / 180.0) <= 5e-25) || ~(((angle_m / 180.0) <= 5e+56)))
		tmp = 2.0 * ((a_m + b) * ((b - a_m) * sin((pi * (angle_m * 0.005555555555555556)))));
	else
		tmp = ((a_m + b) * (b - a_m)) * sin(((angle_m * pi) * 0.011111111111111112));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[Or[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-25], N[Not[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+56]], $MachinePrecision]], N[(2.0 * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999962e-25 or 5.00000000000000024e56 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. difference-of-squares 58.7%

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

   4. Applied egg-rr 58.7%

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

   5. Taylor expanded in angle around inf 56.8%

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

      1. *-commutative 56.8%

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

      2. *-commutative 56.8%

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

      3. associate-*r* 57.4%

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

      4. *-commutative 57.4%

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

      5. associate-*l* 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in angle around inf 66.0%

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

      1. *-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-*r* 67.8%

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

   10. Simplified 67.8%

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

   11. Taylor expanded in angle around 0 67.5%

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

   if 4.99999999999999962e-25 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000024e56

   1. Initial program 52.3%

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

      1. associate-*l* 52.3%

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

      2. *-commutative 52.3%

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

      3. associate-*l* 52.3%

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

   3. Simplified 52.3%

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

      1. *-commutative 52.3%

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

      2. sub-neg 52.3%

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

      3. distribute-lft-in 52.3%

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

      4. 2-sin 52.3%

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

      5. div-inv 52.3%

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

      6. metadata-eval 52.3%

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

   6. Applied egg-rr 52.3%

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

      1. distribute-lft-out 52.3%

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

      2. sub-neg 52.3%

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

      3. *-commutative 52.3%

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

      4. *-commutative 52.3%

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

      5. associate-*r* 58.3%

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

      6. *-commutative 58.3%

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

      7. associate-*l* 58.3%

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

      8. metadata-eval 58.3%

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

   8. Simplified 58.3%

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

      1. unpow2 52.3%

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

      2. unpow2 52.3%

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

      3. difference-of-squares 62.8%

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

   10. Applied egg-rr 68.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-25} \lor \neg \left(\frac{angle}{180} \leq 5 \cdot 10^{+56}\right):\\ \;\;\;\;2 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a\_m + b\right) \cdot \left(b - a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;t\_0 \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a_m b) (- b a_m))))
   (*
    angle_s
    (if (<= b 1.45e+179)
      (* t_0 (sin (* (* angle_m PI) 0.011111111111111112)))
      (* 0.011111111111111112 (* angle_m (* PI t_0)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (a_m + b) * (b - a_m);
	double tmp;
	if (b <= 1.45e+179) {
		tmp = t_0 * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * t_0));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (a_m + b) * (b - a_m);
	double tmp;
	if (b <= 1.45e+179) {
		tmp = t_0 * Math.sin(((angle_m * Math.PI) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = (a_m + b) * (b - a_m)
	tmp = 0
	if b <= 1.45e+179:
		tmp = t_0 * math.sin(((angle_m * math.pi) * 0.011111111111111112))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * t_0))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(a_m + b) * Float64(b - a_m))
	tmp = 0.0
	if (b <= 1.45e+179)
		tmp = Float64(t_0 * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = (a_m + b) * (b - a_m);
	tmp = 0.0;
	if (b <= 1.45e+179)
		tmp = t_0 * sin(((angle_m * pi) * 0.011111111111111112));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 1.45e+179], N[(t$95$0 * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45000000000000009e179

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-*l* 58.6%

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

   3. Simplified 58.6%

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

      1. *-commutative 58.6%

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

      2. sub-neg 58.6%

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

      3. distribute-lft-in 58.6%

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

      4. 2-sin 58.6%

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

      5. div-inv 58.0%

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

      6. metadata-eval 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. distribute-lft-out 58.0%

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

      2. sub-neg 58.0%

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

      3. *-commutative 58.0%

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

      4. *-commutative 58.0%

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

      5. associate-*r* 57.1%

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

      6. *-commutative 57.1%

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

      7. associate-*l* 57.1%

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

      8. metadata-eval 57.1%

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

   8. Simplified 57.1%

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

      1. unpow2 58.6%

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

      2. unpow2 58.6%

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

      3. difference-of-squares 60.4%

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

   10. Applied egg-rr 58.9%

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

   if 1.45000000000000009e179 < b

   1. Initial program 36.8%

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

      1. associate-*l* 36.8%

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

      2. *-commutative 36.8%

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

      3. associate-*l* 36.8%

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

   3. Simplified 36.8%

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

      1. unpow2 36.8%

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

      2. unpow2 36.8%

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

      3. difference-of-squares 47.7%

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

   6. Applied egg-rr 47.7%

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

   7. Taylor expanded in angle around 0 58.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+179}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.8% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. associate-*l* 56.2%

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

   2. *-commutative 56.2%

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

   3. associate-*l* 56.2%

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

3. Simplified 56.2%

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

   1. unpow2 56.2%

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

   2. unpow2 56.2%

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

   3. difference-of-squares 59.0%

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

6. Applied egg-rr 59.0%

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

7. Taylor expanded in angle around 0 55.9%

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

8. Final simplification 55.9%

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

Alternative 13: 54.8% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. associate-*l* 56.2%

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

   2. *-commutative 56.2%

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

   3. associate-*l* 56.2%

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

3. Simplified 56.2%

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

   1. unpow2 56.2%

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

   2. unpow2 56.2%

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

   3. difference-of-squares 59.0%

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

6. Applied egg-rr 59.0%

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

7. Taylor expanded in angle around 0 55.9%

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

   1. *-commutative 55.9%

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

   2. *-commutative 55.9%

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

   3. associate-*r* 55.9%

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

   4. +-commutative 55.9%

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

9. Simplified 55.9%

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

10. Final simplification 55.9%

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

Reproduce

herbie shell --seed 2024112 
(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)))))