ab-angle->ABCF B

Percentage Accurate: 53.6% → 60.6%

Time: 16.6s

Alternatives: 13

Speedup: 32.2×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% 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: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ t_1 := \sqrt{\pi \cdot 0.005555555555555556}\\ t_2 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_3 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(t\_0 \cdot \sin \left(\frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot t\_3\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+164}:\\ \;\;\;\;t\_3 \cdot \left(t\_0 \cdot \sin \left(t\_1 \cdot \left(angle\_m \cdot t\_1\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;\left(t\_0 \cdot \sin \left(e^{\log \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\cos t\_2}^{2}} \cdot \left(t\_0 \cdot {\left(\sqrt[3]{\sin t\_2}\right)}^{3}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ a b) (- b a))))
        (t_1 (sqrt (* PI 0.005555555555555556)))
        (t_2 (* PI (* angle_m 0.005555555555555556)))
        (t_3 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-61)
      (+
       (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
       (*
        b
        (+
         (* 0.011111111111111112 (* angle_m (* PI b)))
         (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
      (if (<= (/ angle_m 180.0) 2e+83)
        (*
         (* t_0 (sin (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0)))))
         t_3)
        (if (<= (/ angle_m 180.0) 5e+164)
          (* t_3 (* t_0 (sin (* t_1 (* angle_m t_1)))))
          (if (<= (/ angle_m 180.0) 5e+190)
            (*
             (* t_0 (sin (exp (log (* angle_m (* PI 0.005555555555555556))))))
             (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))
            (*
             (sqrt (pow (cos t_2) 2.0))
             (* t_0 (pow (cbrt (sin t_2)) 3.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * ((a + b) * (b - a));
	double t_1 = sqrt((((double) M_PI) * 0.005555555555555556));
	double t_2 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_3 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else if ((angle_m / 180.0) <= 2e+83) {
		tmp = (t_0 * sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))) * t_3;
	} else if ((angle_m / 180.0) <= 5e+164) {
		tmp = t_3 * (t_0 * sin((t_1 * (angle_m * t_1))));
	} else if ((angle_m / 180.0) <= 5e+190) {
		tmp = (t_0 * sin(exp(log((angle_m * (((double) M_PI) * 0.005555555555555556)))))) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0)));
	} else {
		tmp = sqrt(pow(cos(t_2), 2.0)) * (t_0 * pow(cbrt(sin(t_2)), 3.0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * ((a + b) * (b - a));
	double t_1 = Math.sqrt((Math.PI * 0.005555555555555556));
	double t_2 = Math.PI * (angle_m * 0.005555555555555556);
	double t_3 = Math.cos(((angle_m / 180.0) * Math.PI));
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else if ((angle_m / 180.0) <= 2e+83) {
		tmp = (t_0 * Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))) * t_3;
	} else if ((angle_m / 180.0) <= 5e+164) {
		tmp = t_3 * (t_0 * Math.sin((t_1 * (angle_m * t_1))));
	} else if ((angle_m / 180.0) <= 5e+190) {
		tmp = (t_0 * Math.sin(Math.exp(Math.log((angle_m * (Math.PI * 0.005555555555555556)))))) * Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0)));
	} else {
		tmp = Math.sqrt(Math.pow(Math.cos(t_2), 2.0)) * (t_0 * Math.pow(Math.cbrt(Math.sin(t_2)), 3.0));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(a + b) * Float64(b - a)))
	t_1 = sqrt(Float64(pi * 0.005555555555555556))
	t_2 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_3 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-61)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	elseif (Float64(angle_m / 180.0) <= 2e+83)
		tmp = Float64(Float64(t_0 * sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))) * t_3);
	elseif (Float64(angle_m / 180.0) <= 5e+164)
		tmp = Float64(t_3 * Float64(t_0 * sin(Float64(t_1 * Float64(angle_m * t_1)))));
	elseif (Float64(angle_m / 180.0) <= 5e+190)
		tmp = Float64(Float64(t_0 * sin(exp(log(Float64(angle_m * Float64(pi * 0.005555555555555556)))))) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0))));
	else
		tmp = Float64(sqrt((cos(t_2) ^ 2.0)) * Float64(t_0 * (cbrt(sin(t_2)) ^ 3.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-61], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+83], N[(N[(t$95$0 * N[Sin[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] * t$95$3), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+164], N[(t$95$3 * N[(t$95$0 * N[Sin[N[(t$95$1 * N[(angle$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+190], N[(N[(t$95$0 * N[Sin[N[Exp[N[Log[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[N[Cos[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Power[N[Power[N[Sin[t$95$2], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-61

   1. Initial program 59.8%

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

   3. Taylor expanded in angle around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. difference-of-squares 62.3%

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

   5. Applied egg-rr 62.3%

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

   6. Taylor expanded in b around 0 65.8%

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

   if 4.0000000000000002e-61 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000006e83

   1. Initial program 58.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 62.4%

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

   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. Step-by-step derivation

      1. add-cube-cbrt 75.9%

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

      2. pow2 75.9%

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

   6. Applied egg-rr 75.9%

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

   if 2.00000000000000006e83 < (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999995e164

   1. Initial program 51.1%

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

      1. unpow2 23.2%

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

      2. unpow2 23.2%

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

      3. difference-of-squares 23.2%

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

   4. Applied egg-rr 55.5%

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

      1. *-commutative 55.5%

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

      2. div-inv 45.7%

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

      3. metadata-eval 45.7%

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

      4. *-commutative 45.7%

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

      5. associate-*r* 45.0%

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

      6. add-exp-log 58.6%

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

      7. *-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. associate-*r* 58.6%

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

      10. *-commutative 58.6%

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

      11. associate-*r* 58.6%

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

   6. Applied egg-rr 58.6%

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

      1. rem-exp-log 53.4%

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

      2. *-commutative 53.4%

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

      3. add-sqr-sqrt 43.2%

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

      4. associate-*r* 61.6%

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

   8. Applied egg-rr 61.6%

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

   if 4.9999999999999995e164 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000036e190

   1. Initial program 25.2%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 26.1%

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

   4. Applied egg-rr 25.2%

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

      1. *-commutative 25.2%

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

      2. div-inv 25.2%

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

      3. metadata-eval 25.2%

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

      4. *-commutative 25.2%

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

      5. associate-*r* 25.2%

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

      6. add-exp-log 50.2%

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

      7. *-commutative 50.2%

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

      8. *-commutative 50.2%

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

      9. associate-*r* 50.2%

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

      10. *-commutative 50.2%

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

      11. associate-*r* 50.2%

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

   6. Applied egg-rr 50.2%

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

      1. add-sqr-sqrt 79.2%

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

      2. pow2 79.2%

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

   8. Applied egg-rr 79.2%

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

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

   1. Initial program 40.3%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. difference-of-squares 41.3%

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

   4. Applied egg-rr 47.0%

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

      1. add-cube-cbrt 47.0%

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

      2. pow3 47.0%

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

      3. div-inv 46.6%

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

      4. metadata-eval 46.6%

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

   6. Applied egg-rr 46.6%

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

      1. add-sqr-sqrt 42.9%

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

      2. sqrt-unprod 58.8%

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

      3. pow2 58.8%

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

      4. div-inv 58.8%

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

      5. metadata-eval 58.8%

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

   8. Applied egg-rr 58.8%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right) \cdot \color{blue}{\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+164}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt{\pi \cdot 0.005555555555555556} \cdot \left(angle \cdot \sqrt{\pi \cdot 0.005555555555555556}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(e^{\log \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ t_1 := \sqrt{\pi \cdot 0.005555555555555556}\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(t\_0 \cdot \sin \left(\frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot t\_2\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+164}:\\ \;\;\;\;t\_2 \cdot \left(t\_0 \cdot \sin \left(t\_1 \cdot \left(angle\_m \cdot t\_1\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;\left(t\_0 \cdot \sin \left(e^{\log \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ a b) (- b a))))
        (t_1 (sqrt (* PI 0.005555555555555556)))
        (t_2 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-61)
      (+
       (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
       (*
        b
        (+
         (* 0.011111111111111112 (* angle_m (* PI b)))
         (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
      (if (<= (/ angle_m 180.0) 2e+83)
        (*
         (* t_0 (sin (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0)))))
         t_2)
        (if (<= (/ angle_m 180.0) 5e+164)
          (* t_2 (* t_0 (sin (* t_1 (* angle_m t_1)))))
          (if (<= (/ angle_m 180.0) 5e+190)
            (*
             (* t_0 (sin (exp (log (* angle_m (* PI 0.005555555555555556))))))
             (cos (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))
            (*
             t_0
             (pow
              (cbrt (sin (* PI (* angle_m 0.005555555555555556))))
              3.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * ((a + b) * (b - a));
	double t_1 = sqrt((((double) M_PI) * 0.005555555555555556));
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else if ((angle_m / 180.0) <= 2e+83) {
		tmp = (t_0 * sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))) * t_2;
	} else if ((angle_m / 180.0) <= 5e+164) {
		tmp = t_2 * (t_0 * sin((t_1 * (angle_m * t_1))));
	} else if ((angle_m / 180.0) <= 5e+190) {
		tmp = (t_0 * sin(exp(log((angle_m * (((double) M_PI) * 0.005555555555555556)))))) * cos(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0)));
	} else {
		tmp = t_0 * pow(cbrt(sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), 3.0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * ((a + b) * (b - a));
	double t_1 = Math.sqrt((Math.PI * 0.005555555555555556));
	double t_2 = Math.cos(((angle_m / 180.0) * Math.PI));
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else if ((angle_m / 180.0) <= 2e+83) {
		tmp = (t_0 * Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))) * t_2;
	} else if ((angle_m / 180.0) <= 5e+164) {
		tmp = t_2 * (t_0 * Math.sin((t_1 * (angle_m * t_1))));
	} else if ((angle_m / 180.0) <= 5e+190) {
		tmp = (t_0 * Math.sin(Math.exp(Math.log((angle_m * (Math.PI * 0.005555555555555556)))))) * Math.cos(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0)));
	} else {
		tmp = t_0 * Math.pow(Math.cbrt(Math.sin((Math.PI * (angle_m * 0.005555555555555556)))), 3.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(a + b) * Float64(b - a)))
	t_1 = sqrt(Float64(pi * 0.005555555555555556))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-61)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	elseif (Float64(angle_m / 180.0) <= 2e+83)
		tmp = Float64(Float64(t_0 * sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))) * t_2);
	elseif (Float64(angle_m / 180.0) <= 5e+164)
		tmp = Float64(t_2 * Float64(t_0 * sin(Float64(t_1 * Float64(angle_m * t_1)))));
	elseif (Float64(angle_m / 180.0) <= 5e+190)
		tmp = Float64(Float64(t_0 * sin(exp(log(Float64(angle_m * Float64(pi * 0.005555555555555556)))))) * cos(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0))));
	else
		tmp = Float64(t_0 * (cbrt(sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 3.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-61], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+83], N[(N[(t$95$0 * N[Sin[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] * t$95$2), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+164], N[(t$95$2 * N[(t$95$0 * N[Sin[N[(t$95$1 * N[(angle$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+190], N[(N[(t$95$0 * N[Sin[N[Exp[N[Log[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Power[N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-61

   1. Initial program 59.8%

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

   3. Taylor expanded in angle around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. difference-of-squares 62.3%

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

   5. Applied egg-rr 62.3%

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

   6. Taylor expanded in b around 0 65.8%

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

   if 4.0000000000000002e-61 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000006e83

   1. Initial program 58.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 62.4%

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

   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. Step-by-step derivation

      1. add-cube-cbrt 75.9%

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

      2. pow2 75.9%

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

   6. Applied egg-rr 75.9%

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

   if 2.00000000000000006e83 < (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999995e164

   1. Initial program 51.1%

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

      1. unpow2 23.2%

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

      2. unpow2 23.2%

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

      3. difference-of-squares 23.2%

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

   4. Applied egg-rr 55.5%

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

      1. *-commutative 55.5%

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

      2. div-inv 45.7%

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

      3. metadata-eval 45.7%

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

      4. *-commutative 45.7%

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

      5. associate-*r* 45.0%

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

      6. add-exp-log 58.6%

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

      7. *-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. associate-*r* 58.6%

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

      10. *-commutative 58.6%

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

      11. associate-*r* 58.6%

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

   6. Applied egg-rr 58.6%

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

      1. rem-exp-log 53.4%

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

      2. *-commutative 53.4%

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

      3. add-sqr-sqrt 43.2%

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

      4. associate-*r* 61.6%

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

   8. Applied egg-rr 61.6%

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

   if 4.9999999999999995e164 < (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000036e190

   1. Initial program 25.2%

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

      1. unpow2 26.1%

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

      2. unpow2 26.1%

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

      3. difference-of-squares 26.1%

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

   4. Applied egg-rr 25.2%

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

      1. *-commutative 25.2%

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

      2. div-inv 25.2%

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

      3. metadata-eval 25.2%

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

      4. *-commutative 25.2%

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

      5. associate-*r* 25.2%

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

      6. add-exp-log 50.2%

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

      7. *-commutative 50.2%

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

      8. *-commutative 50.2%

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

      9. associate-*r* 50.2%

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

      10. *-commutative 50.2%

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

      11. associate-*r* 50.2%

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

   6. Applied egg-rr 50.2%

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

      1. add-sqr-sqrt 79.2%

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

      2. pow2 79.2%

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

   8. Applied egg-rr 79.2%

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

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

   1. Initial program 40.3%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. difference-of-squares 41.3%

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

   4. Applied egg-rr 47.0%

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

      1. add-cube-cbrt 47.0%

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

      2. pow3 47.0%

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

      3. div-inv 46.6%

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

      4. metadata-eval 46.6%

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

   6. Applied egg-rr 46.6%

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

   7. Taylor expanded in angle around 0 58.3%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right) \cdot \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+164}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt{\pi \cdot 0.005555555555555556} \cdot \left(angle \cdot \sqrt{\pi \cdot 0.005555555555555556}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+190}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(e^{\log \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ t_1 := \sqrt{\pi \cdot 0.005555555555555556}\\ t_2 := \frac{angle\_m}{180} \cdot \pi\\ t_3 := \cos t\_2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(t\_0 \cdot \sin \left(\frac{angle\_m}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot t\_3\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+164}:\\ \;\;\;\;t\_3 \cdot \left(t\_0 \cdot \sin \left(t\_1 \cdot \left(angle\_m \cdot t\_1\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\left(t\_0 \cdot \sin t\_2\right) \cdot \cos \left({\left(\sqrt[3]{angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (* (+ a b) (- b a))))
        (t_1 (sqrt (* PI 0.005555555555555556)))
        (t_2 (* (/ angle_m 180.0) PI))
        (t_3 (cos t_2)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-61)
      (+
       (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
       (*
        b
        (+
         (* 0.011111111111111112 (* angle_m (* PI b)))
         (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
      (if (<= (/ angle_m 180.0) 2e+83)
        (*
         (* t_0 (sin (* (/ angle_m 180.0) (* (cbrt PI) (pow (cbrt PI) 2.0)))))
         t_3)
        (if (<= (/ angle_m 180.0) 5e+164)
          (* t_3 (* t_0 (sin (* t_1 (* angle_m t_1)))))
          (if (<= (/ angle_m 180.0) 2e+209)
            (*
             (* t_0 (sin t_2))
             (cos (pow (cbrt (* angle_m (* PI 0.005555555555555556))) 3.0)))
            (*
             t_0
             (pow
              (cbrt (sin (* PI (* angle_m 0.005555555555555556))))
              3.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * ((a + b) * (b - a));
	double t_1 = sqrt((((double) M_PI) * 0.005555555555555556));
	double t_2 = (angle_m / 180.0) * ((double) M_PI);
	double t_3 = cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else if ((angle_m / 180.0) <= 2e+83) {
		tmp = (t_0 * sin(((angle_m / 180.0) * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))) * t_3;
	} else if ((angle_m / 180.0) <= 5e+164) {
		tmp = t_3 * (t_0 * sin((t_1 * (angle_m * t_1))));
	} else if ((angle_m / 180.0) <= 2e+209) {
		tmp = (t_0 * sin(t_2)) * cos(pow(cbrt((angle_m * (((double) M_PI) * 0.005555555555555556))), 3.0));
	} else {
		tmp = t_0 * pow(cbrt(sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), 3.0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * ((a + b) * (b - a));
	double t_1 = Math.sqrt((Math.PI * 0.005555555555555556));
	double t_2 = (angle_m / 180.0) * Math.PI;
	double t_3 = Math.cos(t_2);
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else if ((angle_m / 180.0) <= 2e+83) {
		tmp = (t_0 * Math.sin(((angle_m / 180.0) * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))) * t_3;
	} else if ((angle_m / 180.0) <= 5e+164) {
		tmp = t_3 * (t_0 * Math.sin((t_1 * (angle_m * t_1))));
	} else if ((angle_m / 180.0) <= 2e+209) {
		tmp = (t_0 * Math.sin(t_2)) * Math.cos(Math.pow(Math.cbrt((angle_m * (Math.PI * 0.005555555555555556))), 3.0));
	} else {
		tmp = t_0 * Math.pow(Math.cbrt(Math.sin((Math.PI * (angle_m * 0.005555555555555556)))), 3.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64(Float64(a + b) * Float64(b - a)))
	t_1 = sqrt(Float64(pi * 0.005555555555555556))
	t_2 = Float64(Float64(angle_m / 180.0) * pi)
	t_3 = cos(t_2)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-61)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	elseif (Float64(angle_m / 180.0) <= 2e+83)
		tmp = Float64(Float64(t_0 * sin(Float64(Float64(angle_m / 180.0) * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0))))) * t_3);
	elseif (Float64(angle_m / 180.0) <= 5e+164)
		tmp = Float64(t_3 * Float64(t_0 * sin(Float64(t_1 * Float64(angle_m * t_1)))));
	elseif (Float64(angle_m / 180.0) <= 2e+209)
		tmp = Float64(Float64(t_0 * sin(t_2)) * cos((cbrt(Float64(angle_m * Float64(pi * 0.005555555555555556))) ^ 3.0)));
	else
		tmp = Float64(t_0 * (cbrt(sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 3.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(Pi * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-61], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+83], N[(N[(t$95$0 * N[Sin[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] * t$95$3), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+164], N[(t$95$3 * N[(t$95$0 * N[Sin[N[(t$95$1 * N[(angle$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+209], N[(N[(t$95$0 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[Power[N[Power[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Power[N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-61

   1. Initial program 59.8%

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

   3. Taylor expanded in angle around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. difference-of-squares 62.3%

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

   5. Applied egg-rr 62.3%

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

   6. Taylor expanded in b around 0 65.8%

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

   if 4.0000000000000002e-61 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000006e83

   1. Initial program 58.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. difference-of-squares 62.4%

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

   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. Step-by-step derivation

      1. add-cube-cbrt 75.9%

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

      2. pow2 75.9%

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

   6. Applied egg-rr 75.9%

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

   if 2.00000000000000006e83 < (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999995e164

   1. Initial program 51.1%

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

      1. unpow2 23.2%

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

      2. unpow2 23.2%

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

      3. difference-of-squares 23.2%

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

   4. Applied egg-rr 55.5%

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

      1. *-commutative 55.5%

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

      2. div-inv 45.7%

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

      3. metadata-eval 45.7%

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

      4. *-commutative 45.7%

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

      5. associate-*r* 45.0%

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

      6. add-exp-log 58.6%

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

      7. *-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. associate-*r* 58.6%

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

      10. *-commutative 58.6%

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

      11. associate-*r* 58.6%

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

   6. Applied egg-rr 58.6%

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

      1. rem-exp-log 53.4%

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

      2. *-commutative 53.4%

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

      3. add-sqr-sqrt 43.2%

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

      4. associate-*r* 61.6%

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

   8. Applied egg-rr 61.6%

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

   if 4.9999999999999995e164 < (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000001e209

   1. Initial program 26.8%

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

      1. unpow2 26.5%

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

      2. unpow2 26.5%

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

      3. difference-of-squares 34.8%

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

   4. Applied egg-rr 35.1%

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

      1. div-inv 35.3%

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

      2. metadata-eval 35.3%

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

      3. add-cube-cbrt 71.2%

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

      4. pow3 69.9%

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

      5. *-commutative 69.9%

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

      6. associate-*r* 69.4%

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

   6. Applied egg-rr 69.4%

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

   if 2.0000000000000001e209 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 45.0%

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

      1. unpow2 33.0%

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

      2. unpow2 33.0%

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

      3. difference-of-squares 42.1%

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

   4. Applied egg-rr 49.5%

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

      1. add-cube-cbrt 49.5%

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

      2. pow3 49.5%

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

      3. div-inv 49.5%

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

      4. metadata-eval 49.5%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in angle around 0 59.4%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right) \cdot \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+164}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt{\pi \cdot 0.005555555555555556} \cdot \left(angle \cdot \sqrt{\pi \cdot 0.005555555555555556}\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+209}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a + b\right) \cdot \left(b - a\right)\\ t_1 := \frac{angle\_m}{180} \cdot \pi\\ t_2 := \cos t\_1\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-92}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+154}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(t\_2 \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+182}:\\ \;\;\;\;t\_2 \cdot \left(\sin t\_1 \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left|b - a\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot t\_0\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a b) (- b a)))
        (t_1 (* (/ angle_m 180.0) PI))
        (t_2 (cos t_1)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e-92)
      (+
       (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
       (*
        b
        (+
         (* 0.011111111111111112 (* angle_m (* PI b)))
         (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
      (if (<= (/ angle_m 180.0) 1e+154)
        (* t_0 (* 2.0 (* t_2 (sin (* angle_m (* PI 0.005555555555555556))))))
        (if (<= (/ angle_m 180.0) 1e+182)
          (* t_2 (* (sin t_1) (* 2.0 (* (+ a b) (fabs (- b a))))))
          (*
           (* 2.0 t_0)
           (pow
            (cbrt (sin (* PI (* angle_m 0.005555555555555556))))
            3.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double t_1 = (angle_m / 180.0) * ((double) M_PI);
	double t_2 = cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 1e-92) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else if ((angle_m / 180.0) <= 1e+154) {
		tmp = t_0 * (2.0 * (t_2 * sin((angle_m * (((double) M_PI) * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 1e+182) {
		tmp = t_2 * (sin(t_1) * (2.0 * ((a + b) * fabs((b - a)))));
	} else {
		tmp = (2.0 * t_0) * pow(cbrt(sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), 3.0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double t_1 = (angle_m / 180.0) * Math.PI;
	double t_2 = Math.cos(t_1);
	double tmp;
	if ((angle_m / 180.0) <= 1e-92) {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else if ((angle_m / 180.0) <= 1e+154) {
		tmp = t_0 * (2.0 * (t_2 * Math.sin((angle_m * (Math.PI * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 1e+182) {
		tmp = t_2 * (Math.sin(t_1) * (2.0 * ((a + b) * Math.abs((b - a)))));
	} else {
		tmp = (2.0 * t_0) * Math.pow(Math.cbrt(Math.sin((Math.PI * (angle_m * 0.005555555555555556)))), 3.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(a + b) * Float64(b - a))
	t_1 = Float64(Float64(angle_m / 180.0) * pi)
	t_2 = cos(t_1)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-92)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	elseif (Float64(angle_m / 180.0) <= 1e+154)
		tmp = Float64(t_0 * Float64(2.0 * Float64(t_2 * sin(Float64(angle_m * Float64(pi * 0.005555555555555556))))));
	elseif (Float64(angle_m / 180.0) <= 1e+182)
		tmp = Float64(t_2 * Float64(sin(t_1) * Float64(2.0 * Float64(Float64(a + b) * abs(Float64(b - a))))));
	else
		tmp = Float64(Float64(2.0 * t_0) * (cbrt(sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 3.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-92], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+154], N[(t$95$0 * N[(2.0 * N[(t$95$2 * N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+182], N[(t$95$2 * N[(N[Sin[t$95$1], $MachinePrecision] * N[(2.0 * N[(N[(a + b), $MachinePrecision] * N[Abs[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * t$95$0), $MachinePrecision] * N[Power[N[Power[N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999988e-93

   1. Initial program 58.1%

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

   3. Taylor expanded in angle around 0 58.2%

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

      1. unpow2 58.2%

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

      2. unpow2 58.2%

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

      3. difference-of-squares 60.8%

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

   5. Applied egg-rr 60.8%

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

   6. Taylor expanded in b around 0 64.5%

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

   if 9.99999999999999988e-93 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000004e154

   1. Initial program 61.6%

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

      1. associate-*l* 61.5%

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

      2. *-commutative 61.5%

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

      3. associate-*l* 61.5%

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

   3. Simplified 61.5%

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

      1. unpow2 50.7%

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

      2. unpow2 50.7%

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

      3. difference-of-squares 54.3%

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

   6. Applied egg-rr 67.0%

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

   7. Taylor expanded in angle around inf 64.9%

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

      1. associate-*r* 64.7%

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

      2. *-commutative 64.7%

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

      3. associate-*r* 68.5%

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

   9. Simplified 68.5%

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

   if 1.00000000000000004e154 < (/.f64 angle #s(literal 180 binary64)) < 1.0000000000000001e182

   1. Initial program 38.3%

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

      1. unpow2 17.3%

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

      2. unpow2 17.3%

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

      3. difference-of-squares 17.3%

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

   4. Applied egg-rr 38.3%

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

      1. add-sqr-sqrt 16.7%

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

      2. sqrt-unprod 53.9%

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

      3. pow2 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. unpow2 53.9%

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

      2. rem-sqrt-square 53.9%

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

   8. Simplified 53.9%

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

   if 1.0000000000000001e182 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 39.7%

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

      1. unpow2 31.7%

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

      2. unpow2 31.7%

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

      3. difference-of-squares 40.8%

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

   4. Applied egg-rr 45.8%

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

      1. add-cube-cbrt 45.8%

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

      2. pow3 45.8%

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

      3. div-inv 45.4%

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

      4. metadata-eval 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in angle around 0 56.5%

      \[\leadsto \left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right) \cdot \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-92}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+154}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+182}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left|b - a\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot {\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(2 \cdot \left(angle\_m \cdot \frac{\pi}{180}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e-61)
    (+
     (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
     (*
      b
      (+
       (* 0.011111111111111112 (* angle_m (* PI b)))
       (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
    (*
     (* (+ a b) (- b a))
     (log1p (expm1 (sin (* 2.0 (* angle_m (/ PI 180.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else {
		tmp = ((a + b) * (b - a)) * log1p(expm1(sin((2.0 * (angle_m * (((double) M_PI) / 180.0))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else {
		tmp = ((a + b) * (b - a)) * Math.log1p(Math.expm1(Math.sin((2.0 * (angle_m * (Math.PI / 180.0))))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e-61:
		tmp = (-0.011111111111111112 * (math.pow(a, 2.0) * (angle_m * math.pi))) + (b * ((0.011111111111111112 * (angle_m * (math.pi * b))) + (0.011111111111111112 * (angle_m * (math.pi * (a - a))))))
	else:
		tmp = ((a + b) * (b - a)) * math.log1p(math.expm1(math.sin((2.0 * (angle_m * (math.pi / 180.0))))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-61)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * log1p(expm1(sin(Float64(2.0 * Float64(angle_m * Float64(pi / 180.0)))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-61], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Log[1 + N[(Exp[N[Sin[N[(2.0 * N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-61

   1. Initial program 59.8%

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

   3. Taylor expanded in angle around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. difference-of-squares 62.3%

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

   5. Applied egg-rr 62.3%

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

   6. Taylor expanded in b around 0 65.8%

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

   if 4.0000000000000002e-61 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

      1. unpow2 36.8%

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

      2. unpow2 36.8%

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

      3. difference-of-squares 42.7%

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

   6. Applied egg-rr 54.3%

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

   8. Applied egg-rr 54.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(2 \cdot \left(angle\_m \cdot \frac{\pi}{180}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e-61)
    (+
     (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
     (*
      b
      (+
       (* 0.011111111111111112 (* angle_m (* PI b)))
       (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
    (*
     (* (+ a b) (- b a))
     (expm1 (log1p (sin (* 2.0 (* angle_m (/ PI 180.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else {
		tmp = ((a + b) * (b - a)) * expm1(log1p(sin((2.0 * (angle_m * (((double) M_PI) / 180.0))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else {
		tmp = ((a + b) * (b - a)) * Math.expm1(Math.log1p(Math.sin((2.0 * (angle_m * (Math.PI / 180.0))))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e-61:
		tmp = (-0.011111111111111112 * (math.pow(a, 2.0) * (angle_m * math.pi))) + (b * ((0.011111111111111112 * (angle_m * (math.pi * b))) + (0.011111111111111112 * (angle_m * (math.pi * (a - a))))))
	else:
		tmp = ((a + b) * (b - a)) * math.expm1(math.log1p(math.sin((2.0 * (angle_m * (math.pi / 180.0))))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-61)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * expm1(log1p(sin(Float64(2.0 * Float64(angle_m * Float64(pi / 180.0)))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-61], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[Log[1 + N[Sin[N[(2.0 * N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-61

   1. Initial program 59.8%

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

   3. Taylor expanded in angle around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. difference-of-squares 62.3%

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

   5. Applied egg-rr 62.3%

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

   6. Taylor expanded in b around 0 65.8%

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

   if 4.0000000000000002e-61 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

      1. unpow2 36.8%

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

      2. unpow2 36.8%

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

      3. difference-of-squares 42.7%

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

   6. Applied egg-rr 54.3%

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

   8. Applied egg-rr 54.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle\_m \cdot \frac{\pi}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e-61)
    (+
     (* -0.011111111111111112 (* (pow a 2.0) (* angle_m PI)))
     (*
      b
      (+
       (* 0.011111111111111112 (* angle_m (* PI b)))
       (* 0.011111111111111112 (* angle_m (* PI (- a a)))))))
    (* (* (+ a b) (- b a)) (sin (* 2.0 (* angle_m (/ PI 180.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (pow(a, 2.0) * (angle_m * ((double) M_PI)))) + (b * ((0.011111111111111112 * (angle_m * (((double) M_PI) * b))) + (0.011111111111111112 * (angle_m * (((double) M_PI) * (a - a))))));
	} else {
		tmp = ((a + b) * (b - a)) * sin((2.0 * (angle_m * (((double) M_PI) / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-61) {
		tmp = (-0.011111111111111112 * (Math.pow(a, 2.0) * (angle_m * Math.PI))) + (b * ((0.011111111111111112 * (angle_m * (Math.PI * b))) + (0.011111111111111112 * (angle_m * (Math.PI * (a - a))))));
	} else {
		tmp = ((a + b) * (b - a)) * Math.sin((2.0 * (angle_m * (Math.PI / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e-61:
		tmp = (-0.011111111111111112 * (math.pow(a, 2.0) * (angle_m * math.pi))) + (b * ((0.011111111111111112 * (angle_m * (math.pi * b))) + (0.011111111111111112 * (angle_m * (math.pi * (a - a))))))
	else:
		tmp = ((a + b) * (b - a)) * math.sin((2.0 * (angle_m * (math.pi / 180.0))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-61)
		tmp = Float64(Float64(-0.011111111111111112 * Float64((a ^ 2.0) * Float64(angle_m * pi))) + Float64(b * Float64(Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * b))) + Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a - a)))))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * sin(Float64(2.0 * Float64(angle_m * Float64(pi / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-61)
		tmp = (-0.011111111111111112 * ((a ^ 2.0) * (angle_m * pi))) + (b * ((0.011111111111111112 * (angle_m * (pi * b))) + (0.011111111111111112 * (angle_m * (pi * (a - a))))));
	else
		tmp = ((a + b) * (b - a)) * sin((2.0 * (angle_m * (pi / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-61], N[(N[(-0.011111111111111112 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e-61

   1. Initial program 59.8%

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

   3. Taylor expanded in angle around 0 59.9%

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

      1. unpow2 59.9%

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

      2. unpow2 59.9%

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

      3. difference-of-squares 62.3%

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

   5. Applied egg-rr 62.3%

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

   6. Taylor expanded in b around 0 65.8%

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

   if 4.0000000000000002e-61 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

      1. unpow2 36.8%

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

      2. unpow2 36.8%

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

      3. difference-of-squares 42.7%

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

   6. Applied egg-rr 54.3%

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

      1. 2-sin 54.3%

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

      2. clear-num 54.9%

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

      3. div-inv 53.7%

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

      4. associate-/r/ 54.1%

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

   8. Applied egg-rr 54.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-61}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) + b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) + 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-92}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right) - {a}^{2} \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle\_m \cdot \frac{\pi}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e-92)
    (*
     0.011111111111111112
     (- (* b (* angle_m (* PI b))) (* (pow a 2.0) (* angle_m PI))))
    (* (* (+ a b) (- b a)) (sin (* 2.0 (* angle_m (/ PI 180.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e-92) {
		tmp = 0.011111111111111112 * ((b * (angle_m * (((double) M_PI) * b))) - (pow(a, 2.0) * (angle_m * ((double) M_PI))));
	} else {
		tmp = ((a + b) * (b - a)) * sin((2.0 * (angle_m * (((double) M_PI) / 180.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e-92) {
		tmp = 0.011111111111111112 * ((b * (angle_m * (Math.PI * b))) - (Math.pow(a, 2.0) * (angle_m * Math.PI)));
	} else {
		tmp = ((a + b) * (b - a)) * Math.sin((2.0 * (angle_m * (Math.PI / 180.0))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e-92:
		tmp = 0.011111111111111112 * ((b * (angle_m * (math.pi * b))) - (math.pow(a, 2.0) * (angle_m * math.pi)))
	else:
		tmp = ((a + b) * (b - a)) * math.sin((2.0 * (angle_m * (math.pi / 180.0))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-92)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * Float64(angle_m * Float64(pi * b))) - Float64((a ^ 2.0) * Float64(angle_m * pi))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * sin(Float64(2.0 * Float64(angle_m * Float64(pi / 180.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e-92)
		tmp = 0.011111111111111112 * ((b * (angle_m * (pi * b))) - ((a ^ 2.0) * (angle_m * pi)));
	else
		tmp = ((a + b) * (b - a)) * sin((2.0 * (angle_m * (pi / 180.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-92], N[(0.011111111111111112 * N[(N[(b * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 2.0], $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999988e-93

   1. Initial program 58.1%

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

   3. Taylor expanded in angle around 0 58.2%

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

      1. unpow2 58.2%

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

      2. unpow2 58.2%

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

      3. difference-of-squares 60.8%

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

   5. Applied egg-rr 60.8%

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

   6. Taylor expanded in b around 0 64.5%

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

      1. +-commutative 64.5%

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

      2. mul-1-neg 64.5%

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

      3. unsub-neg 64.5%

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

      4. distribute-lft-out 64.5%

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

      5. *-commutative 64.5%

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

      6. distribute-rgt1-in 64.5%

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

      7. metadata-eval 64.5%

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

      8. mul0-lft 64.5%

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

      9. distribute-rgt-out 64.5%

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

   8. Simplified 64.5%

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

   if 9.99999999999999988e-93 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 52.4%

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

      1. associate-*l* 52.4%

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

      2. *-commutative 52.4%

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

      3. associate-*l* 52.4%

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

   3. Simplified 52.4%

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

      1. unpow2 41.9%

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

      2. unpow2 41.9%

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

      3. difference-of-squares 47.2%

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

   6. Applied egg-rr 57.7%

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

      1. 2-sin 57.7%

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

      2. clear-num 58.3%

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

      3. div-inv 57.2%

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

      4. associate-/r/ 57.5%

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

   8. Applied egg-rr 57.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-92}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot b\right)\right) - {a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a + b\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 7.4 \cdot 10^{+171}:\\ \;\;\;\;t\_0 \cdot \sin \left(2 \cdot \left(\frac{angle\_m}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a b) (- b a))))
   (*
    angle_s
    (if (<= a 7.4e+171)
      (* t_0 (sin (* 2.0 (* (/ angle_m 180.0) PI))))
      (* 0.011111111111111112 (* angle_m (* PI t_0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double tmp;
	if (a <= 7.4e+171) {
		tmp = t_0 * sin((2.0 * ((angle_m / 180.0) * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * t_0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (a + b) * (b - a);
	double tmp;
	if (a <= 7.4e+171) {
		tmp = t_0 * Math.sin((2.0 * ((angle_m / 180.0) * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * t_0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (a + b) * (b - a)
	tmp = 0
	if a <= 7.4e+171:
		tmp = t_0 * math.sin((2.0 * ((angle_m / 180.0) * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * t_0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(a + b) * Float64(b - a))
	tmp = 0.0
	if (a <= 7.4e+171)
		tmp = Float64(t_0 * sin(Float64(2.0 * Float64(Float64(angle_m / 180.0) * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (a + b) * (b - a);
	tmp = 0.0;
	if (a <= 7.4e+171)
		tmp = t_0 * sin((2.0 * ((angle_m / 180.0) * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 7.4e+171], N[(t$95$0 * N[Sin[N[(2.0 * N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]), $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 a < 7.39999999999999996e171

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. *-commutative 57.0%

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

      3. associate-*l* 57.0%

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

   3. Simplified 57.0%

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

      1. unpow2 51.5%

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

      2. unpow2 51.5%

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

      3. difference-of-squares 53.7%

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

   6. Applied egg-rr 59.2%

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

   8. Applied egg-rr 60.8%

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

      1. unpow1 60.8%

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

      2. associate-*l/ 59.3%

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

      3. associate-/l* 59.2%

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

   10. Simplified 59.2%

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

   if 7.39999999999999996e171 < a

   1. Initial program 46.3%

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

   3. Taylor expanded in angle around 0 58.8%

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

      1. unpow2 58.8%

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

      2. unpow2 58.8%

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

      3. difference-of-squares 75.9%

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

   5. Applied egg-rr 75.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.4 \cdot 10^{+171}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. associate-*l* 56.0%

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

   2. *-commutative 56.0%

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

   3. associate-*l* 56.0%

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

3. Simplified 56.0%

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

   1. unpow2 52.2%

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

   2. unpow2 52.2%

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

   3. difference-of-squares 55.8%

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

6. Applied egg-rr 59.6%

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

   1. 2-sin 59.6%

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

   2. clear-num 60.9%

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

   3. div-inv 60.7%

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

   4. associate-/r/ 61.8%

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

8. Applied egg-rr 61.8%

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

9. Final simplification 61.8%

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

Alternative 11: 53.8% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. associate-*l* 56.0%

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

   2. *-commutative 56.0%

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

   3. associate-*l* 56.0%

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

3. Simplified 56.0%

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

   1. unpow2 52.2%

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

   2. unpow2 52.2%

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

   3. difference-of-squares 55.8%

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

6. Applied egg-rr 59.6%

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

7. Taylor expanded in angle around 0 55.8%

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

   1. *-commutative 55.8%

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

9. Simplified 55.8%

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

10. Final simplification 55.8%

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

Alternative 12: 53.9% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. unpow2 52.2%

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

   2. unpow2 52.2%

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

   3. difference-of-squares 55.8%

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

4. Applied egg-rr 59.6%

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

   1. div-inv 59.2%

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

   2. metadata-eval 59.2%

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

   3. add-sqr-sqrt 30.0%

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

   4. pow2 30.0%

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

   5. *-commutative 30.0%

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

   6. associate-*r* 29.7%

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

6. Applied egg-rr 29.7%

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

7. Taylor expanded in angle around 0 55.8%

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

   1. *-commutative 55.8%

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

   2. *-commutative 55.8%

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

   3. +-commutative 55.8%

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

   4. associate-*l* 55.8%

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

   5. +-commutative 55.8%

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

   6. *-commutative 55.8%

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

   7. *-commutative 55.8%

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

   8. associate-*r* 55.8%

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

   9. *-commutative 55.8%

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

9. Simplified 55.8%

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

10. Final simplification 55.8%

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

Alternative 13: 53.8% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in angle around 0 52.2%

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

   1. unpow2 52.2%

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

   2. unpow2 52.2%

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

   3. difference-of-squares 55.8%

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

5. Applied egg-rr 55.8%

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

6. Final simplification 55.8%

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

Reproduce

herbie shell --seed 2024132 
(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)))))