ab-angle->ABCF B

Percentage Accurate: 53.8% → 67.3%

Time: 16.8s

Alternatives: 21

Speedup: 23.3×

• 32.5% 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: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, angle\_m \cdot 0.005555555555555556, 1\right)\\ t_1 := \left(a + b\right) \cdot \left(b - a\right)\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ t_3 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_4 := \sin t\_3\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\cos t\_0, \cos 1, \sin t\_0 \cdot \sin 1\right)\right) \cdot \left(\left(t\_4 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+186}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\sqrt{{t\_4}^{2}} \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_2 \cdot \sin \left({\left(\sqrt[3]{t\_3}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (fma PI (* angle_m 0.005555555555555556) 1.0))
        (t_1 (* (+ a b) (- b a)))
        (t_2 (cos (* (/ angle_m 180.0) PI)))
        (t_3 (* PI (* angle_m 0.005555555555555556)))
        (t_4 (sin t_3)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+36)
      (*
       (* 2.0 (fma (cos t_0) (cos 1.0) (* (sin t_0) (sin 1.0))))
       (* (* t_4 (+ a b)) (- b a)))
      (if (<= (/ angle_m 180.0) 5e+186)
        (* t_1 (* 2.0 (* (sqrt (pow t_4 2.0)) t_2)))
        (* t_1 (* 2.0 (* t_2 (sin (pow (cbrt t_3) 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 = fma(((double) M_PI), (angle_m * 0.005555555555555556), 1.0);
	double t_1 = (a + b) * (b - a);
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double t_3 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_4 = sin(t_3);
	double tmp;
	if ((angle_m / 180.0) <= 2e+36) {
		tmp = (2.0 * fma(cos(t_0), cos(1.0), (sin(t_0) * sin(1.0)))) * ((t_4 * (a + b)) * (b - a));
	} else if ((angle_m / 180.0) <= 5e+186) {
		tmp = t_1 * (2.0 * (sqrt(pow(t_4, 2.0)) * t_2));
	} else {
		tmp = t_1 * (2.0 * (t_2 * sin(pow(cbrt(t_3), 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 = fma(pi, Float64(angle_m * 0.005555555555555556), 1.0)
	t_1 = Float64(Float64(a + b) * Float64(b - a))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	t_3 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_4 = sin(t_3)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+36)
		tmp = Float64(Float64(2.0 * fma(cos(t_0), cos(1.0), Float64(sin(t_0) * sin(1.0)))) * Float64(Float64(t_4 * Float64(a + b)) * Float64(b - a)));
	elseif (Float64(angle_m / 180.0) <= 5e+186)
		tmp = Float64(t_1 * Float64(2.0 * Float64(sqrt((t_4 ^ 2.0)) * t_2)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_2 * sin((cbrt(t_3) ^ 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[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+36], N[(N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[1.0], $MachinePrecision] + N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+186], N[(t$95$1 * N[(2.0 * N[(N[Sqrt[N[Power[t$95$4, 2.0], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(t$95$2 * N[Sin[N[Power[N[Power[t$95$3, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000008e36

   1. Initial program 61.3%

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

      1. associate-*l* 61.3%

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

      2. *-commutative 61.3%

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

      3. associate-*l* 61.3%

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

   3. Simplified 61.3%

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

      1. unpow2 61.3%

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

      2. unpow2 61.3%

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

      3. difference-of-squares 64.9%

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

   6. Applied egg-rr 64.9%

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

   7. Taylor expanded in angle around inf 65.1%

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

      1. associate-*r* 65.2%

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

      2. *-commutative 65.2%

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

      3. *-commutative 65.2%

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

      4. associate-*r* 66.1%

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

      5. associate-*r* 76.0%

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

      6. *-commutative 76.0%

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

      7. *-commutative 76.0%

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

      8. associate-*r* 73.6%

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

   9. Simplified 73.6%

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

       1. metadata-eval 73.6%

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

       2. div-inv 74.2%

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

       3. expm1-log1p-u 64.1%

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

       4. expm1-undefine 63.6%

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

       5. *-commutative 63.6%

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

       6. div-inv 63.6%

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

       7. metadata-eval 63.6%

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

       8. associate-*r* 63.6%

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

       9. *-commutative 63.6%

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

   11. Applied egg-rr 63.6%

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

       1. expm1-define 64.1%

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

   13. Simplified 64.1%

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

       1. expm1-undefine 63.6%

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

       2. cos-diff 64.1%

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

       3. log1p-expm1-u 64.1%

          \[\leadsto \left(2 \cdot \left(\cos \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       4. log1p-undefine 64.1%

          \[\leadsto \left(2 \cdot \left(\cos \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. rem-exp-log 63.6%

          \[\leadsto \left(2 \cdot \left(\cos \color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin 1\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. expm1-log1p-u 63.6%

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

       7. log1p-expm1-u 63.6%

          \[\leadsto \left(2 \cdot \left(\cos \left(1 + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \cos 1 + \sin \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}}\right) \cdot \sin 1\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       8. log1p-undefine 63.6%

          \[\leadsto \left(2 \cdot \left(\cos \left(1 + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \cos 1 + \sin \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}}\right) \cdot \sin 1\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       9. rem-exp-log 64.1%

          \[\leadsto \left(2 \cdot \left(\cos \left(1 + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \cos 1 + \sin \color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \sin 1\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       10. expm1-log1p-u 76.5%

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

   15. Applied egg-rr 76.5%

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

       1. fma-define 76.4%

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

       2. +-commutative 76.4%

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

       3. associate-*r* 76.9%

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

       4. *-commutative 76.9%

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

       5. associate-*r* 76.4%

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

       6. fma-define 76.9%

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

       7. +-commutative 76.9%

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

       8. associate-*r* 77.7%

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

       9. *-commutative 77.7%

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

       10. associate-*r* 78.8%

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

       11. fma-define 78.8%

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

   17. Simplified 78.8%

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

   if 2.00000000000000008e36 < (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999954e186

   1. Initial program 26.6%

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

      1. associate-*l* 26.6%

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

      2. *-commutative 26.6%

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

      3. associate-*l* 26.6%

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

   3. Simplified 26.6%

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

      1. unpow2 26.6%

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

      2. unpow2 26.6%

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

      3. difference-of-squares 29.5%

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

   6. Applied egg-rr 29.5%

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

      1. add-sqr-sqrt 18.7%

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

      2. sqrt-unprod 43.6%

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

      3. pow2 43.6%

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

      4. div-inv 43.5%

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

      5. metadata-eval 43.5%

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

   8. Applied egg-rr 43.5%

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

   if 4.99999999999999954e186 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. *-commutative 24.8%

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

      3. associate-*l* 24.8%

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

   3. Simplified 24.8%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   6. Applied egg-rr 24.8%

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

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

   8. Applied egg-rr 28.2%

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

      1. unpow2 28.2%

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

      2. add-sqr-sqrt 24.8%

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

      3. add-cube-cbrt 37.4%

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

      4. pow3 41.4%

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

      5. div-inv 45.9%

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

      6. metadata-eval 45.9%

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

   10. Applied egg-rr 45.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\left(2 \cdot \mathsf{fma}\left(\cos \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right), \cos 1, \sin \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right) \cdot \sin 1\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \left(a + b\right) \cdot \left(b - a\right)\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ t_3 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\left(\left(t\_3 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot {\left({\left(angle\_m \cdot 0.005555555555555556\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+186}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\sqrt{{t\_3}^{2}} \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_2 \cdot \sin \left({\left(\sqrt[3]{t\_0}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* (+ a b) (- b a)))
        (t_2 (cos (* (/ angle_m 180.0) PI)))
        (t_3 (sin t_0)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+98)
      (*
       (* (* t_3 (+ a b)) (- b a))
       (*
        2.0
        (cos
         (*
          PI
          (pow
           (pow (* angle_m 0.005555555555555556) 3.0)
           0.3333333333333333)))))
      (if (<= (/ angle_m 180.0) 5e+186)
        (* t_1 (* 2.0 (* (sqrt (pow t_3 2.0)) t_2)))
        (* t_1 (* 2.0 (* t_2 (sin (pow (cbrt t_0) 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 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = (a + b) * (b - a);
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double t_3 = sin(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 5e+98) {
		tmp = ((t_3 * (a + b)) * (b - a)) * (2.0 * cos((((double) M_PI) * pow(pow((angle_m * 0.005555555555555556), 3.0), 0.3333333333333333))));
	} else if ((angle_m / 180.0) <= 5e+186) {
		tmp = t_1 * (2.0 * (sqrt(pow(t_3, 2.0)) * t_2));
	} else {
		tmp = t_1 * (2.0 * (t_2 * sin(pow(cbrt(t_0), 3.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = (a + b) * (b - a);
	double t_2 = Math.cos(((angle_m / 180.0) * Math.PI));
	double t_3 = Math.sin(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 5e+98) {
		tmp = ((t_3 * (a + b)) * (b - a)) * (2.0 * Math.cos((Math.PI * Math.pow(Math.pow((angle_m * 0.005555555555555556), 3.0), 0.3333333333333333))));
	} else if ((angle_m / 180.0) <= 5e+186) {
		tmp = t_1 * (2.0 * (Math.sqrt(Math.pow(t_3, 2.0)) * t_2));
	} else {
		tmp = t_1 * (2.0 * (t_2 * Math.sin(Math.pow(Math.cbrt(t_0), 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(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(Float64(a + b) * Float64(b - a))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	t_3 = sin(t_0)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+98)
		tmp = Float64(Float64(Float64(t_3 * Float64(a + b)) * Float64(b - a)) * Float64(2.0 * cos(Float64(pi * ((Float64(angle_m * 0.005555555555555556) ^ 3.0) ^ 0.3333333333333333)))));
	elseif (Float64(angle_m / 180.0) <= 5e+186)
		tmp = Float64(t_1 * Float64(2.0 * Float64(sqrt((t_3 ^ 2.0)) * t_2)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_2 * sin((cbrt(t_0) ^ 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[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+98], N[(N[(N[(t$95$3 * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[N[(Pi * N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+186], N[(t$95$1 * N[(2.0 * N[(N[Sqrt[N[Power[t$95$3, 2.0], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(t$95$2 * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.9999999999999998e98

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. *-commutative 58.5%

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

      3. associate-*l* 58.5%

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

   3. Simplified 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. difference-of-squares 62.2%

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

   6. Applied egg-rr 62.2%

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

   7. Taylor expanded in angle around inf 61.9%

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

      1. associate-*r* 61.9%

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

      2. *-commutative 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-*r* 62.2%

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

      5. associate-*r* 71.3%

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

      6. *-commutative 71.3%

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

      7. *-commutative 71.3%

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

      8. associate-*r* 70.1%

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

   9. Simplified 70.1%

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

       1. metadata-eval 70.1%

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

       2. div-inv 70.7%

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

       3. add-cbrt-cube 65.5%

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

       4. pow1/3 48.8%

          \[\leadsto \left(2 \cdot \cos \left(\pi \cdot \color{blue}{{\left(\left(\frac{angle}{180} \cdot \frac{angle}{180}\right) \cdot \frac{angle}{180}\right)}^{0.3333333333333333}}\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       5. pow3 48.8%

          \[\leadsto \left(2 \cdot \cos \left(\pi \cdot {\color{blue}{\left({\left(\frac{angle}{180}\right)}^{3}\right)}}^{0.3333333333333333}\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       6. div-inv 48.7%

          \[\leadsto \left(2 \cdot \cos \left(\pi \cdot {\left({\color{blue}{\left(angle \cdot \frac{1}{180}\right)}}^{3}\right)}^{0.3333333333333333}\right)\right) \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       7. metadata-eval 48.7%

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

   11. Applied egg-rr 48.7%

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

   if 4.9999999999999998e98 < (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999954e186

   1. Initial program 26.0%

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

      1. associate-*l* 26.0%

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

      2. *-commutative 26.0%

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

      3. associate-*l* 26.0%

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

   3. Simplified 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

      3. difference-of-squares 26.0%

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

   6. Applied egg-rr 26.0%

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

      1. add-sqr-sqrt 13.6%

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

      2. sqrt-unprod 44.4%

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

      3. pow2 44.4%

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

      4. div-inv 44.4%

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

      5. metadata-eval 44.4%

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

   8. Applied egg-rr 44.4%

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

   if 4.99999999999999954e186 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. *-commutative 24.8%

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

      3. associate-*l* 24.8%

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

   3. Simplified 24.8%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   6. Applied egg-rr 24.8%

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

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

   8. Applied egg-rr 28.2%

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

      1. unpow2 28.2%

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

      2. add-sqr-sqrt 24.8%

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

      3. add-cube-cbrt 37.4%

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

      4. pow3 41.4%

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

      5. div-inv 45.9%

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

      6. metadata-eval 45.9%

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

   10. Applied egg-rr 45.9%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+98}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot {\left({\left(angle \cdot 0.005555555555555556\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ t_2 := \sin t\_0\\ t_3 := \left(a + b\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(t\_2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+186}:\\ \;\;\;\;t\_3 \cdot \left(2 \cdot \left(\sqrt{{t\_2}^{2}} \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(2 \cdot \left(t\_1 \cdot \sin \left({\left(\sqrt[3]{t\_0}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (cos (* (/ angle_m 180.0) PI)))
        (t_2 (sin t_0))
        (t_3 (* (+ a b) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+20)
      (* (* (* t_2 (+ a b)) (- b a)) (* 2.0 (cos t_0)))
      (if (<= (/ angle_m 180.0) 5e+186)
        (* t_3 (* 2.0 (* (sqrt (pow t_2 2.0)) t_1)))
        (* t_3 (* 2.0 (* t_1 (sin (pow (cbrt t_0) 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 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double t_2 = sin(t_0);
	double t_3 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((t_2 * (a + b)) * (b - a)) * (2.0 * cos(t_0));
	} else if ((angle_m / 180.0) <= 5e+186) {
		tmp = t_3 * (2.0 * (sqrt(pow(t_2, 2.0)) * t_1));
	} else {
		tmp = t_3 * (2.0 * (t_1 * sin(pow(cbrt(t_0), 3.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = Math.cos(((angle_m / 180.0) * Math.PI));
	double t_2 = Math.sin(t_0);
	double t_3 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((t_2 * (a + b)) * (b - a)) * (2.0 * Math.cos(t_0));
	} else if ((angle_m / 180.0) <= 5e+186) {
		tmp = t_3 * (2.0 * (Math.sqrt(Math.pow(t_2, 2.0)) * t_1));
	} else {
		tmp = t_3 * (2.0 * (t_1 * Math.sin(Math.pow(Math.cbrt(t_0), 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(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = cos(Float64(Float64(angle_m / 180.0) * pi))
	t_2 = sin(t_0)
	t_3 = Float64(Float64(a + b) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+20)
		tmp = Float64(Float64(Float64(t_2 * Float64(a + b)) * Float64(b - a)) * Float64(2.0 * cos(t_0)));
	elseif (Float64(angle_m / 180.0) <= 5e+186)
		tmp = Float64(t_3 * Float64(2.0 * Float64(sqrt((t_2 ^ 2.0)) * t_1)));
	else
		tmp = Float64(t_3 * Float64(2.0 * Float64(t_1 * sin((cbrt(t_0) ^ 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[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+20], N[(N[(N[(t$95$2 * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+186], N[(t$95$3 * N[(2.0 * N[(N[Sqrt[N[Power[t$95$2, 2.0], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(2.0 * N[(t$95$1 * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e20

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in angle around inf 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r* 66.7%

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

      5. associate-*r* 76.7%

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

      6. *-commutative 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-*r* 74.4%

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

   9. Simplified 74.4%

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

   if 4e20 < (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999954e186

   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. Step-by-step derivation

      1. associate-*l* 25.2%

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

      2. *-commutative 25.2%

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

      3. associate-*l* 25.2%

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

   3. Simplified 25.2%

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

      1. unpow2 25.2%

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

      2. unpow2 25.2%

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

      3. difference-of-squares 27.9%

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

   6. Applied egg-rr 27.9%

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

      1. add-sqr-sqrt 17.7%

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

      2. sqrt-unprod 43.9%

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

      3. pow2 43.9%

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

      4. div-inv 43.9%

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

      5. metadata-eval 43.9%

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

   8. Applied egg-rr 43.9%

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

   if 4.99999999999999954e186 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. *-commutative 24.8%

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

      3. associate-*l* 24.8%

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

   3. Simplified 24.8%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   6. Applied egg-rr 24.8%

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

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

   8. Applied egg-rr 28.2%

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

      1. unpow2 28.2%

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

      2. add-sqr-sqrt 24.8%

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

      3. add-cube-cbrt 37.4%

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

      4. pow3 41.4%

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

      5. div-inv 45.9%

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

      6. metadata-eval 45.9%

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

   10. Applied egg-rr 45.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}} \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \left(a + b\right) \cdot \left(b - a\right)\\ t_2 := \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+24}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \sqrt[3]{{\left(angle\_m \cdot 0.005555555555555556\right)}^{3}}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+172}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_2 \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_2 \cdot \sin \left({\left(\sqrt[3]{t\_0}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* (+ a b) (- b a)))
        (t_2 (cos (* (/ angle_m 180.0) PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+24)
      (*
       (* (* (sin t_0) (+ a b)) (- b a))
       (* 2.0 (cos (* PI (cbrt (pow (* angle_m 0.005555555555555556) 3.0))))))
      (if (<= (/ angle_m 180.0) 2e+172)
        (* t_1 (* 2.0 (* t_2 (* angle_m (* PI 0.005555555555555556)))))
        (* t_1 (* 2.0 (* t_2 (sin (pow (cbrt t_0) 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 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = (a + b) * (b - a);
	double t_2 = cos(((angle_m / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle_m / 180.0) <= 1e+24) {
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * (2.0 * cos((((double) M_PI) * cbrt(pow((angle_m * 0.005555555555555556), 3.0)))));
	} else if ((angle_m / 180.0) <= 2e+172) {
		tmp = t_1 * (2.0 * (t_2 * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = t_1 * (2.0 * (t_2 * sin(pow(cbrt(t_0), 3.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = (a + b) * (b - a);
	double t_2 = Math.cos(((angle_m / 180.0) * Math.PI));
	double tmp;
	if ((angle_m / 180.0) <= 1e+24) {
		tmp = ((Math.sin(t_0) * (a + b)) * (b - a)) * (2.0 * Math.cos((Math.PI * Math.cbrt(Math.pow((angle_m * 0.005555555555555556), 3.0)))));
	} else if ((angle_m / 180.0) <= 2e+172) {
		tmp = t_1 * (2.0 * (t_2 * (angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = t_1 * (2.0 * (t_2 * Math.sin(Math.pow(Math.cbrt(t_0), 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(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(Float64(a + b) * Float64(b - a))
	t_2 = cos(Float64(Float64(angle_m / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+24)
		tmp = Float64(Float64(Float64(sin(t_0) * Float64(a + b)) * Float64(b - a)) * Float64(2.0 * cos(Float64(pi * cbrt((Float64(angle_m * 0.005555555555555556) ^ 3.0))))));
	elseif (Float64(angle_m / 180.0) <= 2e+172)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_2 * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_2 * sin((cbrt(t_0) ^ 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[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $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], 1e+24], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[N[(Pi * N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+172], N[(t$95$1 * N[(2.0 * N[(t$95$2 * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(t$95$2 * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999998e23

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in angle around inf 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r* 66.7%

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

      5. associate-*r* 76.7%

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

      6. *-commutative 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-*r* 74.4%

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

   9. Simplified 74.4%

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

       1. metadata-eval 74.4%

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

       2. div-inv 75.0%

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

       3. add-cbrt-cube 69.5%

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

       4. pow3 68.9%

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

       5. div-inv 69.4%

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

       6. metadata-eval 69.4%

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

   11. Applied egg-rr 69.4%

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

   if 9.9999999999999998e23 < (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000002e172

   1. Initial program 26.7%

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

      1. associate-*l* 26.7%

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

      2. *-commutative 26.7%

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

      3. associate-*l* 26.7%

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

   3. Simplified 26.7%

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

      1. unpow2 26.7%

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

      2. unpow2 26.7%

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

      3. difference-of-squares 29.6%

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

   6. Applied egg-rr 29.6%

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

   7. Taylor expanded in angle around 0 44.0%

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

      1. associate-*r* 44.0%

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

      2. *-commutative 44.0%

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

      3. associate-*r* 44.0%

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

   9. Simplified 44.0%

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

   if 2.0000000000000002e172 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 22.8%

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

      1. associate-*l* 22.8%

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

      2. *-commutative 22.8%

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

      3. associate-*l* 22.8%

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

   3. Simplified 22.8%

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

      1. unpow2 22.8%

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

      2. unpow2 22.8%

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

      3. difference-of-squares 22.8%

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

   6. Applied egg-rr 22.8%

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

      1. add-sqr-sqrt 25.8%

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

      2. pow2 25.8%

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

   8. Applied egg-rr 25.8%

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

      1. unpow2 25.8%

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

      2. add-sqr-sqrt 22.8%

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

      3. add-cube-cbrt 33.8%

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

      4. pow3 37.2%

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

      5. div-inv 41.2%

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

      6. metadata-eval 41.2%

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

   10. Applied egg-rr 41.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+24}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \sqrt[3]{{\left(angle \cdot 0.005555555555555556\right)}^{3}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+172}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+24}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \sqrt[3]{{\left(angle\_m \cdot 0.005555555555555556\right)}^{3}}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\pi \cdot e^{\log \left(angle\_m \cdot 0.005555555555555556\right)}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+24)
      (*
       (* (* (sin t_0) (+ a b)) (- b a))
       (* 2.0 (cos (* PI (cbrt (pow (* angle_m 0.005555555555555556) 3.0))))))
      (if (<= (/ angle_m 180.0) 1.5e+186)
        (*
         (* (+ a b) (- b a))
         (*
          2.0
          (*
           (cos (* (/ angle_m 180.0) PI))
           (* angle_m (* PI 0.005555555555555556)))))
        (*
         (* 2.0 (cos t_0))
         (*
          (- b a)
          (*
           (+ a b)
           (sin (* PI (exp (log (* angle_m 0.005555555555555556)))))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+24) {
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * (2.0 * cos((((double) M_PI) * cbrt(pow((angle_m * 0.005555555555555556), 3.0)))));
	} else if ((angle_m / 180.0) <= 1.5e+186) {
		tmp = ((a + b) * (b - a)) * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = (2.0 * cos(t_0)) * ((b - a) * ((a + b) * sin((((double) M_PI) * exp(log((angle_m * 0.005555555555555556)))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 1e+24) {
		tmp = ((Math.sin(t_0) * (a + b)) * (b - a)) * (2.0 * Math.cos((Math.PI * Math.cbrt(Math.pow((angle_m * 0.005555555555555556), 3.0)))));
	} else if ((angle_m / 180.0) <= 1.5e+186) {
		tmp = ((a + b) * (b - a)) * (2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = (2.0 * Math.cos(t_0)) * ((b - a) * ((a + b) * Math.sin((Math.PI * Math.exp(Math.log((angle_m * 0.005555555555555556)))))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+24)
		tmp = Float64(Float64(Float64(sin(t_0) * Float64(a + b)) * Float64(b - a)) * Float64(2.0 * cos(Float64(pi * cbrt((Float64(angle_m * 0.005555555555555556) ^ 3.0))))));
	elseif (Float64(angle_m / 180.0) <= 1.5e+186)
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(b - a) * Float64(Float64(a + b) * sin(Float64(pi * exp(log(Float64(angle_m * 0.005555555555555556))))))));
	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[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+24], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[N[(Pi * N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1.5e+186], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[Sin[N[(Pi * N[Exp[N[Log[N[(angle$95$m * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999998e23

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in angle around inf 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r* 66.7%

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

      5. associate-*r* 76.7%

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

      6. *-commutative 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-*r* 74.4%

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

   9. Simplified 74.4%

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

       1. metadata-eval 74.4%

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

       2. div-inv 75.0%

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

       3. add-cbrt-cube 69.5%

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

       4. pow3 68.9%

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

       5. div-inv 69.4%

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

       6. metadata-eval 69.4%

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

   11. Applied egg-rr 69.4%

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

   if 9.9999999999999998e23 < (/.f64 angle #s(literal 180 binary64)) < 1.49999999999999991e186

   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. Step-by-step derivation

      1. associate-*l* 25.2%

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

      2. *-commutative 25.2%

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

      3. associate-*l* 25.2%

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

   3. Simplified 25.2%

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

      1. unpow2 25.2%

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

      2. unpow2 25.2%

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

      3. difference-of-squares 27.9%

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

   6. Applied egg-rr 27.9%

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

   7. Taylor expanded in angle around 0 40.9%

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

      1. associate-*r* 40.9%

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

      2. *-commutative 40.9%

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

      3. associate-*r* 40.9%

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

   9. Simplified 40.9%

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

   if 1.49999999999999991e186 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. *-commutative 24.8%

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

      3. associate-*l* 24.8%

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

   3. Simplified 24.8%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in angle around inf 28.2%

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

      1. associate-*r* 28.2%

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

      2. *-commutative 28.2%

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

      3. *-commutative 28.2%

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

      4. associate-*r* 20.0%

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

      5. associate-*r* 20.0%

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

      6. *-commutative 20.0%

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

      7. *-commutative 20.0%

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

      8. associate-*r* 21.1%

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

   9. Simplified 21.1%

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

       1. add-exp-log 40.4%

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

   11. Applied egg-rr 40.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+24}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \sqrt[3]{{\left(angle \cdot 0.005555555555555556\right)}^{3}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\pi \cdot e^{\log \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := 2 \cdot \cos t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot t\_1\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\pi \cdot e^{\log \left(angle\_m \cdot 0.005555555555555556\right)}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))) (t_1 (* 2.0 (cos t_0))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+20)
      (* (* (* (sin t_0) (+ a b)) (- b a)) t_1)
      (if (<= (/ angle_m 180.0) 1.5e+186)
        (*
         (* (+ a b) (- b a))
         (*
          2.0
          (*
           (cos (* (/ angle_m 180.0) PI))
           (* angle_m (* PI 0.005555555555555556)))))
        (*
         t_1
         (*
          (- b a)
          (*
           (+ a b)
           (sin (* PI (exp (log (* angle_m 0.005555555555555556)))))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * t_1;
	} else if ((angle_m / 180.0) <= 1.5e+186) {
		tmp = ((a + b) * (b - a)) * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = t_1 * ((b - a) * ((a + b) * sin((((double) M_PI) * exp(log((angle_m * 0.005555555555555556)))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = 2.0 * Math.cos(t_0);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((Math.sin(t_0) * (a + b)) * (b - a)) * t_1;
	} else if ((angle_m / 180.0) <= 1.5e+186) {
		tmp = ((a + b) * (b - a)) * (2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = t_1 * ((b - a) * ((a + b) * Math.sin((Math.PI * Math.exp(Math.log((angle_m * 0.005555555555555556)))))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	t_1 = 2.0 * math.cos(t_0)
	tmp = 0
	if (angle_m / 180.0) <= 4e+20:
		tmp = ((math.sin(t_0) * (a + b)) * (b - a)) * t_1
	elif (angle_m / 180.0) <= 1.5e+186:
		tmp = ((a + b) * (b - a)) * (2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (angle_m * (math.pi * 0.005555555555555556))))
	else:
		tmp = t_1 * ((b - a) * ((a + b) * math.sin((math.pi * math.exp(math.log((angle_m * 0.005555555555555556)))))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(2.0 * cos(t_0))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+20)
		tmp = Float64(Float64(Float64(sin(t_0) * Float64(a + b)) * Float64(b - a)) * t_1);
	elseif (Float64(angle_m / 180.0) <= 1.5e+186)
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(t_1 * Float64(Float64(b - a) * Float64(Float64(a + b) * sin(Float64(pi * exp(log(Float64(angle_m * 0.005555555555555556))))))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m * 0.005555555555555556);
	t_1 = 2.0 * cos(t_0);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+20)
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * t_1;
	elseif ((angle_m / 180.0) <= 1.5e+186)
		tmp = ((a + b) * (b - a)) * (2.0 * (cos(((angle_m / 180.0) * pi)) * (angle_m * (pi * 0.005555555555555556))));
	else
		tmp = t_1 * ((b - a) * ((a + b) * sin((pi * exp(log((angle_m * 0.005555555555555556)))))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+20], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1.5e+186], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[Sin[N[(Pi * N[Exp[N[Log[N[(angle$95$m * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e20

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in angle around inf 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r* 66.7%

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

      5. associate-*r* 76.7%

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

      6. *-commutative 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-*r* 74.4%

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

   9. Simplified 74.4%

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

   if 4e20 < (/.f64 angle #s(literal 180 binary64)) < 1.49999999999999991e186

   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. Step-by-step derivation

      1. associate-*l* 25.2%

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

      2. *-commutative 25.2%

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

      3. associate-*l* 25.2%

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

   3. Simplified 25.2%

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

      1. unpow2 25.2%

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

      2. unpow2 25.2%

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

      3. difference-of-squares 27.9%

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

   6. Applied egg-rr 27.9%

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

   7. Taylor expanded in angle around 0 40.9%

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

      1. associate-*r* 40.9%

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

      2. *-commutative 40.9%

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

      3. associate-*r* 40.9%

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

   9. Simplified 40.9%

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

   if 1.49999999999999991e186 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.8%

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

      1. associate-*l* 24.8%

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

      2. *-commutative 24.8%

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

      3. associate-*l* 24.8%

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

   3. Simplified 24.8%

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

      1. unpow2 24.8%

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

      2. unpow2 24.8%

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

      3. difference-of-squares 24.8%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in angle around inf 28.2%

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

      1. associate-*r* 28.2%

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

      2. *-commutative 28.2%

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

      3. *-commutative 28.2%

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

      4. associate-*r* 20.0%

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

      5. associate-*r* 20.0%

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

      6. *-commutative 20.0%

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

      7. *-commutative 20.0%

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

      8. associate-*r* 21.1%

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

   9. Simplified 21.1%

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

       1. add-exp-log 40.4%

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

   11. Applied egg-rr 40.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\pi \cdot e^{\log \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \left(a + b\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+166}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\cos \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \sin \left(\frac{angle\_m}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* (+ a b) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+20)
      (* (* (* (sin t_0) (+ a b)) (- b a)) (* 2.0 (cos t_0)))
      (if (<= (/ angle_m 180.0) 5e+166)
        (*
         t_1
         (*
          2.0
          (*
           (cos (* (/ angle_m 180.0) PI))
           (* angle_m (* PI 0.005555555555555556)))))
        (* t_1 (* 2.0 (sin (* (/ angle_m 180.0) (pow (sqrt PI) 2.0))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * (2.0 * cos(t_0));
	} else if ((angle_m / 180.0) <= 5e+166) {
		tmp = t_1 * (2.0 * (cos(((angle_m / 180.0) * ((double) M_PI))) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = t_1 * (2.0 * sin(((angle_m / 180.0) * pow(sqrt(((double) M_PI)), 2.0))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((Math.sin(t_0) * (a + b)) * (b - a)) * (2.0 * Math.cos(t_0));
	} else if ((angle_m / 180.0) <= 5e+166) {
		tmp = t_1 * (2.0 * (Math.cos(((angle_m / 180.0) * Math.PI)) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = t_1 * (2.0 * Math.sin(((angle_m / 180.0) * Math.pow(Math.sqrt(Math.PI), 2.0))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	t_1 = (a + b) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e+20:
		tmp = ((math.sin(t_0) * (a + b)) * (b - a)) * (2.0 * math.cos(t_0))
	elif (angle_m / 180.0) <= 5e+166:
		tmp = t_1 * (2.0 * (math.cos(((angle_m / 180.0) * math.pi)) * (angle_m * (math.pi * 0.005555555555555556))))
	else:
		tmp = t_1 * (2.0 * math.sin(((angle_m / 180.0) * math.pow(math.sqrt(math.pi), 2.0))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(Float64(a + b) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+20)
		tmp = Float64(Float64(Float64(sin(t_0) * Float64(a + b)) * Float64(b - a)) * Float64(2.0 * cos(t_0)));
	elseif (Float64(angle_m / 180.0) <= 5e+166)
		tmp = Float64(t_1 * Float64(2.0 * Float64(cos(Float64(Float64(angle_m / 180.0) * pi)) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(t_1 * Float64(2.0 * sin(Float64(Float64(angle_m / 180.0) * (sqrt(pi) ^ 2.0)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m * 0.005555555555555556);
	t_1 = (a + b) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+20)
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * (2.0 * cos(t_0));
	elseif ((angle_m / 180.0) <= 5e+166)
		tmp = t_1 * (2.0 * (cos(((angle_m / 180.0) * pi)) * (angle_m * (pi * 0.005555555555555556))));
	else
		tmp = t_1 * (2.0 * sin(((angle_m / 180.0) * (sqrt(pi) ^ 2.0))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+20], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+166], N[(t$95$1 * N[(2.0 * N[(N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e20

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in angle around inf 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r* 66.7%

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

      5. associate-*r* 76.7%

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

      6. *-commutative 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-*r* 74.4%

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

   9. Simplified 74.4%

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

   if 4e20 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e166

   1. Initial program 24.5%

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

      1. associate-*l* 24.5%

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

      2. *-commutative 24.5%

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

      3. associate-*l* 24.5%

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

   3. Simplified 24.5%

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

      1. unpow2 24.5%

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

      2. unpow2 24.5%

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

      3. difference-of-squares 27.5%

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

   6. Applied egg-rr 27.5%

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

   7. Taylor expanded in angle around 0 42.4%

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

      1. associate-*r* 42.4%

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

      2. *-commutative 42.4%

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

      3. associate-*r* 42.4%

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

   9. Simplified 42.4%

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

   if 5.0000000000000002e166 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.8%

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

      1. associate-*l* 25.8%

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

      2. *-commutative 25.8%

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

      3. associate-*l* 25.8%

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

   3. Simplified 25.8%

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

      1. unpow2 25.8%

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

      2. unpow2 25.8%

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

      3. difference-of-squares 25.8%

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

   6. Applied egg-rr 25.8%

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

      1. add-sqr-sqrt 28.6%

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

      2. pow2 28.6%

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

   8. Applied egg-rr 28.6%

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

   9. Taylor expanded in angle around 0 36.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+166}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := \frac{angle\_m}{180} \cdot \pi\\ t_2 := \left(a + b\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+222}:\\ \;\;\;\;t\_2 \cdot \left(2 \cdot \left(\cos t\_1 \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(2 \cdot \sin t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556)))
        (t_1 (* (/ angle_m 180.0) PI))
        (t_2 (* (+ a b) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+20)
      (* (* (* (sin t_0) (+ a b)) (- b a)) (* 2.0 (cos t_0)))
      (if (<= (/ angle_m 180.0) 2e+222)
        (* t_2 (* 2.0 (* (cos t_1) (* angle_m (* PI 0.005555555555555556)))))
        (* t_2 (* 2.0 (sin t_1))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = (angle_m / 180.0) * ((double) M_PI);
	double t_2 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * (2.0 * cos(t_0));
	} else if ((angle_m / 180.0) <= 2e+222) {
		tmp = t_2 * (2.0 * (cos(t_1) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = t_2 * (2.0 * sin(t_1));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = (angle_m / 180.0) * Math.PI;
	double t_2 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = ((Math.sin(t_0) * (a + b)) * (b - a)) * (2.0 * Math.cos(t_0));
	} else if ((angle_m / 180.0) <= 2e+222) {
		tmp = t_2 * (2.0 * (Math.cos(t_1) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = t_2 * (2.0 * Math.sin(t_1));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	t_1 = (angle_m / 180.0) * math.pi
	t_2 = (a + b) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e+20:
		tmp = ((math.sin(t_0) * (a + b)) * (b - a)) * (2.0 * math.cos(t_0))
	elif (angle_m / 180.0) <= 2e+222:
		tmp = t_2 * (2.0 * (math.cos(t_1) * (angle_m * (math.pi * 0.005555555555555556))))
	else:
		tmp = t_2 * (2.0 * math.sin(t_1))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(Float64(angle_m / 180.0) * pi)
	t_2 = Float64(Float64(a + b) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+20)
		tmp = Float64(Float64(Float64(sin(t_0) * Float64(a + b)) * Float64(b - a)) * Float64(2.0 * cos(t_0)));
	elseif (Float64(angle_m / 180.0) <= 2e+222)
		tmp = Float64(t_2 * Float64(2.0 * Float64(cos(t_1) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(t_2 * Float64(2.0 * sin(t_1)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m * 0.005555555555555556);
	t_1 = (angle_m / 180.0) * pi;
	t_2 = (a + b) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+20)
		tmp = ((sin(t_0) * (a + b)) * (b - a)) * (2.0 * cos(t_0));
	elseif ((angle_m / 180.0) <= 2e+222)
		tmp = t_2 * (2.0 * (cos(t_1) * (angle_m * (pi * 0.005555555555555556))));
	else
		tmp = t_2 * (2.0 * sin(t_1));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+20], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+222], N[(t$95$2 * N[(2.0 * N[(N[Cos[t$95$1], $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(2.0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e20

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

   7. Taylor expanded in angle around inf 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r* 66.7%

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

      5. associate-*r* 76.7%

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

      6. *-commutative 76.7%

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

      7. *-commutative 76.7%

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

      8. associate-*r* 74.4%

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

   9. Simplified 74.4%

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

   if 4e20 < (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000001e222

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 25.1%

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

      2. unpow2 25.1%

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

      3. difference-of-squares 27.2%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in angle around 0 38.3%

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

      1. associate-*r* 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-*r* 38.3%

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

   9. Simplified 38.3%

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

   if 2.0000000000000001e222 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.9%

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

      1. associate-*l* 24.9%

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

      2. *-commutative 24.9%

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

      3. associate-*l* 24.9%

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

   3. Simplified 24.9%

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

      1. unpow2 24.9%

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

      2. unpow2 24.9%

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

      3. difference-of-squares 24.9%

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

   6. Applied egg-rr 24.9%

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

   7. Taylor expanded in angle around 0 41.5%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b 2.0) 2e+303)
    (* (* (+ a b) (- b a)) (sin (* angle_m (* PI 0.011111111111111112))))
    (* b (* (- b a) (sin (* 0.011111111111111112 (* angle_m PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(b, 2.0) <= 2e+303) {
		tmp = ((a + b) * (b - a)) * sin((angle_m * (((double) M_PI) * 0.011111111111111112)));
	} else {
		tmp = b * ((b - a) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (Math.pow(b, 2.0) <= 2e+303) {
		tmp = ((a + b) * (b - a)) * Math.sin((angle_m * (Math.PI * 0.011111111111111112)));
	} else {
		tmp = b * ((b - a) * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if math.pow(b, 2.0) <= 2e+303:
		tmp = ((a + b) * (b - a)) * math.sin((angle_m * (math.pi * 0.011111111111111112)))
	else:
		tmp = b * ((b - a) * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if ((b ^ 2.0) <= 2e+303)
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112))));
	else
		tmp = Float64(b * Float64(Float64(b - a) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((b ^ 2.0) <= 2e+303)
		tmp = ((a + b) * (b - a)) * sin((angle_m * (pi * 0.011111111111111112)));
	else
		tmp = b * ((b - a) * sin((0.011111111111111112 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 2e+303], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2e303

   1. Initial program 57.5%

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

      1. associate-*l* 57.5%

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

      2. *-commutative 57.5%

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

      3. associate-*l* 57.5%

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

   3. Simplified 57.5%

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

      1. unpow2 57.5%

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

      2. unpow2 57.5%

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

      3. difference-of-squares 57.5%

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

   6. Applied egg-rr 57.5%

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

      1. add-sqr-sqrt 59.6%

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

      2. pow2 59.6%

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

   8. Applied egg-rr 59.6%

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

      1. expm1-log1p-u 57.9%

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

      2. expm1-undefine 28.7%

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

   10. Applied egg-rr 25.5%

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

       1. log1p-undefine 25.5%

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

       2. rem-exp-log 25.5%

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

       3. +-commutative 25.5%

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

       4. associate--l+ 55.0%

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

       5. metadata-eval 55.0%

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

       6. +-rgt-identity 55.0%

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

       7. associate-*r* 55.4%

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

       8. *-commutative 55.4%

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

       9. associate-*r* 56.3%

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

       10. count-2 56.3%

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

       11. *-commutative 56.3%

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

       12. *-commutative 56.3%

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

       13. distribute-lft-out 56.3%

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

       14. distribute-rgt-out 56.3%

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

       15. metadata-eval 56.3%

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

   12. Simplified 56.3%

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

   if 2e303 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 42.8%

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

      1. associate-*l* 42.8%

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

      2. *-commutative 42.8%

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

      3. associate-*l* 42.8%

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

   3. Simplified 42.8%

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

      1. unpow2 42.8%

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

      2. unpow2 42.8%

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

      3. difference-of-squares 53.8%

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in b around inf 55.2%

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

      1. add-sqr-sqrt 51.0%

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

      2. sqrt-unprod 53.7%

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

      3. pow2 53.7%

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

   9. Applied egg-rr 53.7%

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

       1. unpow2 53.7%

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

       2. rem-sqrt-square 53.7%

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

   11. Simplified 53.7%

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

       1. pow1 53.7%

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

       2. add-sqr-sqrt 51.0%

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

       3. fabs-sqr 51.0%

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

       4. add-sqr-sqrt 55.2%

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

       5. 2-sin 55.2%

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

       6. div-inv 55.4%

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

       7. metadata-eval 55.4%

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

   13. Applied egg-rr 55.4%

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

       1. unpow1 55.4%

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

       2. associate-*l* 67.4%

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

       3. associate-*r* 71.2%

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

       4. *-commutative 71.2%

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

       5. associate-*r* 65.8%

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

       6. count-2 65.8%

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

       7. associate-*r* 68.5%

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

       8. *-commutative 68.5%

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

       9. associate-*r* 71.2%

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

       10. *-commutative 71.2%

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

       11. distribute-lft-out 71.2%

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

       12. metadata-eval 71.2%

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

       13. *-commutative 71.2%

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

       14. *-commutative 71.2%

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

   15. Simplified 71.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ t_1 := \left(a + b\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+223}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(\cos t\_0 \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \sin t\_0\right)\\ \end{array} \end{array} \end{array} \]

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 (* (/ angle_m 180.0) PI)) (t_1 (* (+ a b) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+20)
      (*
       (+ a b)
       (* (- b a) (sin (* (* angle_m 0.005555555555555556) (* 2.0 PI)))))
      (if (<= (/ angle_m 180.0) 1e+223)
        (* t_1 (* 2.0 (* (cos t_0) (* angle_m (* PI 0.005555555555555556)))))
        (* t_1 (* 2.0 (sin t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	double t_1 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = (a + b) * ((b - a) * sin(((angle_m * 0.005555555555555556) * (2.0 * ((double) M_PI)))));
	} else if ((angle_m / 180.0) <= 1e+223) {
		tmp = t_1 * (2.0 * (cos(t_0) * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = t_1 * (2.0 * sin(t_0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	double t_1 = (a + b) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = (a + b) * ((b - a) * Math.sin(((angle_m * 0.005555555555555556) * (2.0 * Math.PI))));
	} else if ((angle_m / 180.0) <= 1e+223) {
		tmp = t_1 * (2.0 * (Math.cos(t_0) * (angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = t_1 * (2.0 * Math.sin(t_0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (angle_m / 180.0) * math.pi
	t_1 = (a + b) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e+20:
		tmp = (a + b) * ((b - a) * math.sin(((angle_m * 0.005555555555555556) * (2.0 * math.pi))))
	elif (angle_m / 180.0) <= 1e+223:
		tmp = t_1 * (2.0 * (math.cos(t_0) * (angle_m * (math.pi * 0.005555555555555556))))
	else:
		tmp = t_1 * (2.0 * math.sin(t_0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	t_1 = Float64(Float64(a + b) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+20)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * 0.005555555555555556) * Float64(2.0 * pi)))));
	elseif (Float64(angle_m / 180.0) <= 1e+223)
		tmp = Float64(t_1 * Float64(2.0 * Float64(cos(t_0) * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(t_1 * Float64(2.0 * sin(t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (angle_m / 180.0) * pi;
	t_1 = (a + b) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+20)
		tmp = (a + b) * ((b - a) * sin(((angle_m * 0.005555555555555556) * (2.0 * pi))));
	elseif ((angle_m / 180.0) <= 1e+223)
		tmp = t_1 * (2.0 * (cos(t_0) * (angle_m * (pi * 0.005555555555555556))));
	else
		tmp = t_1 * (2.0 * sin(t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+20], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+223], N[(t$95$1 * N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e20

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

      1. add-sqr-sqrt 68.3%

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

      2. pow2 68.3%

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

   8. Applied egg-rr 68.3%

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

      1. pow1 68.3%

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

   10. Applied egg-rr 64.4%

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

       1. unpow1 64.4%

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

       2. associate-*r* 74.4%

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

       3. associate-*r* 74.4%

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

   12. Simplified 74.4%

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

   if 4e20 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000005e223

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 25.1%

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

      2. unpow2 25.1%

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

      3. difference-of-squares 27.2%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in angle around 0 38.3%

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

      1. associate-*r* 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-*r* 38.3%

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

   9. Simplified 38.3%

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

   if 1.00000000000000005e223 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.9%

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

      1. associate-*l* 24.9%

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

      2. *-commutative 24.9%

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

      3. associate-*l* 24.9%

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

   3. Simplified 24.9%

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

      1. unpow2 24.9%

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

      2. unpow2 24.9%

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

      3. difference-of-squares 24.9%

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

   6. Applied egg-rr 24.9%

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

   7. Taylor expanded in angle around 0 41.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+223}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right)\\ \end{array} \end{array} \]

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+20)
    (*
     (+ a b)
     (* (- b a) (sin (* (* angle_m 0.005555555555555556) (* 2.0 PI)))))
    (* (* (+ a b) (- b a)) (* 2.0 (sin (* (/ angle_m 180.0) PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = (a + b) * ((b - a) * sin(((angle_m * 0.005555555555555556) * (2.0 * ((double) M_PI)))));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * sin(((angle_m / 180.0) * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = (a + b) * ((b - a) * Math.sin(((angle_m * 0.005555555555555556) * (2.0 * Math.PI))));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * Math.sin(((angle_m / 180.0) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e+20:
		tmp = (a + b) * ((b - a) * math.sin(((angle_m * 0.005555555555555556) * (2.0 * math.pi))))
	else:
		tmp = ((a + b) * (b - a)) * (2.0 * math.sin(((angle_m / 180.0) * math.pi)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+20)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * 0.005555555555555556) * Float64(2.0 * pi)))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * sin(Float64(Float64(angle_m / 180.0) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+20)
		tmp = (a + b) * ((b - a) * sin(((angle_m * 0.005555555555555556) * (2.0 * pi))));
	else
		tmp = ((a + b) * (b - a)) * (2.0 * sin(((angle_m / 180.0) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+20], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e20

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

      1. add-sqr-sqrt 68.3%

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

      2. pow2 68.3%

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

   8. Applied egg-rr 68.3%

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

      1. pow1 68.3%

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

   10. Applied egg-rr 64.4%

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

       1. unpow1 64.4%

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

       2. associate-*r* 74.4%

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

       3. associate-*r* 74.4%

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

   12. Simplified 74.4%

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

   if 4e20 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 25.1%

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

      2. unpow2 25.1%

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

      3. difference-of-squares 26.8%

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

   6. Applied egg-rr 26.8%

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

   7. Taylor expanded in angle around 0 35.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

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+20)
    (*
     (+ a b)
     (* (- b a) (sin (* (* angle_m 0.005555555555555556) (* 2.0 PI)))))
    (*
     2.0
     (* (* (sin (* PI (* angle_m 0.005555555555555556))) (+ 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) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = (a + b) * ((b - a) * sin(((angle_m * 0.005555555555555556) * (2.0 * ((double) M_PI)))));
	} else {
		tmp = 2.0 * ((sin((((double) M_PI) * (angle_m * 0.005555555555555556))) * (a + b)) * (b - a));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+20) {
		tmp = (a + b) * ((b - a) * Math.sin(((angle_m * 0.005555555555555556) * (2.0 * Math.PI))));
	} else {
		tmp = 2.0 * ((Math.sin((Math.PI * (angle_m * 0.005555555555555556))) * (a + b)) * (b - a));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e+20:
		tmp = (a + b) * ((b - a) * math.sin(((angle_m * 0.005555555555555556) * (2.0 * math.pi))))
	else:
		tmp = 2.0 * ((math.sin((math.pi * (angle_m * 0.005555555555555556))) * (a + b)) * (b - a))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+20)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * 0.005555555555555556) * Float64(2.0 * pi)))));
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(pi * Float64(angle_m * 0.005555555555555556))) * Float64(a + b)) * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+20)
		tmp = (a + b) * ((b - a) * sin(((angle_m * 0.005555555555555556) * (2.0 * pi))));
	else
		tmp = 2.0 * ((sin((pi * (angle_m * 0.005555555555555556))) * (a + b)) * (b - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+20], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e20

   1. Initial program 61.9%

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

      1. associate-*l* 61.9%

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

      2. *-commutative 61.9%

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

      3. associate-*l* 61.9%

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

   3. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

      3. difference-of-squares 65.6%

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

   6. Applied egg-rr 65.6%

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

      1. add-sqr-sqrt 68.3%

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

      2. pow2 68.3%

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

   8. Applied egg-rr 68.3%

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

      1. pow1 68.3%

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

   10. Applied egg-rr 64.4%

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

       1. unpow1 64.4%

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

       2. associate-*r* 74.4%

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

       3. associate-*r* 74.4%

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

   12. Simplified 74.4%

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

   if 4e20 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. *-commutative 25.1%

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

      3. associate-*l* 25.1%

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

   3. Simplified 25.1%

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

      1. unpow2 25.1%

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

      2. unpow2 25.1%

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

      3. difference-of-squares 26.8%

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

   6. Applied egg-rr 26.8%

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

   7. Taylor expanded in angle around inf 29.3%

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

      1. associate-*r* 29.3%

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

      2. *-commutative 29.3%

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

      3. *-commutative 29.3%

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

      4. associate-*r* 20.7%

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

      5. associate-*r* 20.7%

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

      6. *-commutative 20.7%

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

      7. *-commutative 20.7%

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

      8. associate-*r* 23.1%

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

   9. Simplified 23.1%

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

   10. Taylor expanded in angle around 0 35.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+99}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot t\_0\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+155}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

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 (sin (* 0.011111111111111112 (* angle_m PI)))))
   (*
    angle_s
    (if (<= b 6.8e+99)
      (* (* (+ a b) (- b a)) t_0)
      (if (<= b 1.3e+155)
        (* 0.011111111111111112 (* (* angle_m b) (* PI (- b a))))
        (* b (* (- b a) 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 = sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	double tmp;
	if (b <= 6.8e+99) {
		tmp = ((a + b) * (b - a)) * t_0;
	} else if (b <= 1.3e+155) {
		tmp = 0.011111111111111112 * ((angle_m * b) * (((double) M_PI) * (b - a)));
	} else {
		tmp = b * ((b - a) * t_0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((0.011111111111111112 * (angle_m * Math.PI)));
	double tmp;
	if (b <= 6.8e+99) {
		tmp = ((a + b) * (b - a)) * t_0;
	} else if (b <= 1.3e+155) {
		tmp = 0.011111111111111112 * ((angle_m * b) * (Math.PI * (b - a)));
	} else {
		tmp = b * ((b - a) * t_0);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.sin((0.011111111111111112 * (angle_m * math.pi)))
	tmp = 0
	if b <= 6.8e+99:
		tmp = ((a + b) * (b - a)) * t_0
	elif b <= 1.3e+155:
		tmp = 0.011111111111111112 * ((angle_m * b) * (math.pi * (b - a)))
	else:
		tmp = b * ((b - a) * 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 = sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))
	tmp = 0.0
	if (b <= 6.8e+99)
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * t_0);
	elseif (b <= 1.3e+155)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * b) * Float64(pi * Float64(b - a))));
	else
		tmp = Float64(b * Float64(Float64(b - a) * 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 = sin((0.011111111111111112 * (angle_m * pi)));
	tmp = 0.0;
	if (b <= 6.8e+99)
		tmp = ((a + b) * (b - a)) * t_0;
	elseif (b <= 1.3e+155)
		tmp = 0.011111111111111112 * ((angle_m * b) * (pi * (b - a)));
	else
		tmp = b * ((b - a) * 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[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b, 6.8e+99], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[b, 1.3e+155], N[(0.011111111111111112 * N[(N[(angle$95$m * b), $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if b < 6.79999999999999968e99

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-*l* 57.9%

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

   3. Simplified 57.9%

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

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. difference-of-squares 58.9%

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

   6. Applied egg-rr 58.9%

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

      1. add-sqr-sqrt 59.5%

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

      2. pow2 59.5%

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

   8. Applied egg-rr 59.5%

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

      1. pow1 59.5%

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

   10. Applied egg-rr 57.2%

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

       1. unpow1 57.2%

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

       2. *-commutative 57.2%

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

       3. *-commutative 57.2%

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

       4. associate-*r* 58.4%

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

       5. *-commutative 58.4%

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

       6. associate-*r* 57.2%

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

       7. count-2 57.2%

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

       8. associate-*r* 58.3%

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

       9. *-commutative 58.3%

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

       10. associate-*r* 58.4%

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

       11. *-commutative 58.4%

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

       12. distribute-lft-out 58.4%

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

       13. metadata-eval 58.4%

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

       14. *-commutative 58.4%

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

       15. *-commutative 58.4%

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

   12. Simplified 58.4%

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

   if 6.79999999999999968e99 < b < 1.3000000000000001e155

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. *-commutative 38.4%

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

      3. associate-*l* 38.4%

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

   3. Simplified 38.4%

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

      1. unpow2 38.4%

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

      2. unpow2 38.4%

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

      3. difference-of-squares 38.4%

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

   6. Applied egg-rr 38.4%

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

   7. Taylor expanded in b around inf 48.9%

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

   8. Taylor expanded in angle around 0 41.7%

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

      1. associate-*r* 47.0%

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

   10. Simplified 47.0%

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

   if 1.3000000000000001e155 < b

   1. Initial program 36.8%

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

      1. associate-*l* 36.8%

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

      2. *-commutative 36.8%

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

      3. associate-*l* 36.8%

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

   3. Simplified 36.8%

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

      1. unpow2 36.8%

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

      2. unpow2 36.8%

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

      3. difference-of-squares 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in b around inf 54.8%

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

      1. add-sqr-sqrt 47.1%

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

      2. sqrt-unprod 49.7%

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

      3. pow2 49.7%

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

   9. Applied egg-rr 49.7%

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

       1. unpow2 49.7%

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

       2. rem-sqrt-square 49.7%

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

   11. Simplified 49.7%

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

       1. pow1 49.7%

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

       2. add-sqr-sqrt 47.1%

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

       3. fabs-sqr 47.1%

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

       4. add-sqr-sqrt 54.8%

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

       5. 2-sin 54.8%

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

       6. div-inv 52.2%

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

       7. metadata-eval 52.2%

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

   13. Applied egg-rr 52.2%

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

       1. unpow1 52.2%

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

       2. associate-*l* 62.3%

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

       3. associate-*r* 67.5%

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

       4. *-commutative 67.5%

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

       5. associate-*r* 64.9%

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

       6. count-2 64.9%

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

       7. associate-*r* 64.9%

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

       8. *-commutative 64.9%

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

       9. associate-*r* 67.5%

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

       10. *-commutative 67.5%

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

       11. distribute-lft-out 67.5%

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

       12. metadata-eval 67.5%

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

       13. *-commutative 67.5%

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

       14. *-commutative 67.5%

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

   15. Simplified 67.5%

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

Alternative 14: 56.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{+155}:\\ \;\;\;\;angle\_m \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 1.02e+155)
    (* angle_m (* (* (+ a b) (- b a)) (* PI 0.011111111111111112)))
    (* b (* (- b a) (sin (* 0.011111111111111112 (* angle_m PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.02e+155) {
		tmp = angle_m * (((a + b) * (b - a)) * (((double) M_PI) * 0.011111111111111112));
	} else {
		tmp = b * ((b - a) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.02e+155) {
		tmp = angle_m * (((a + b) * (b - a)) * (Math.PI * 0.011111111111111112));
	} else {
		tmp = b * ((b - a) * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 1.02e+155:
		tmp = angle_m * (((a + b) * (b - a)) * (math.pi * 0.011111111111111112))
	else:
		tmp = b * ((b - a) * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 1.02e+155)
		tmp = Float64(angle_m * Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(pi * 0.011111111111111112)));
	else
		tmp = Float64(b * Float64(Float64(b - a) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.02e+155)
		tmp = angle_m * (((a + b) * (b - a)) * (pi * 0.011111111111111112));
	else
		tmp = b * ((b - a) * sin((0.011111111111111112 * (angle_m * pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 1.02e+155], N[(angle$95$m * N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.02e155

   1. Initial program 56.2%

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

      1. associate-*l* 56.2%

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

      2. *-commutative 56.2%

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

      3. associate-*l* 56.2%

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

   3. Simplified 56.2%

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

      1. unpow2 56.2%

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

      2. unpow2 56.2%

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

      3. difference-of-squares 57.2%

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

   6. Applied egg-rr 57.2%

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

   7. Taylor expanded in angle around 0 36.1%

      \[\leadsto \color{blue}{angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. fma-define 36.1%

         \[\leadsto angle \cdot \color{blue}{\mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]

      2. +-commutative 36.1%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      3. *-commutative 36.1%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}, 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      4. +-commutative 36.1%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      5. distribute-rgt-out 36.1%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right), 2 \cdot \left({angle}^{2} \cdot \color{blue}{\left(\left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \left(-8.573388203017833 \cdot 10^{-8} + -2.8577960676726107 \cdot 10^{-8}\right)\right)}\right)\right) \]

   9. Simplified 36.1%

      \[\leadsto \color{blue}{angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(\left({\pi}^{3} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)} \]

   10. Taylor expanded in angle around 0 54.9%

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

       1. associate-*r* 55.0%

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

   12. Simplified 55.0%

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

   if 1.02e155 < b

   1. Initial program 36.8%

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

      1. associate-*l* 36.8%

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

      2. *-commutative 36.8%

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

      3. associate-*l* 36.8%

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

   3. Simplified 36.8%

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

      1. unpow2 36.8%

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

      2. unpow2 36.8%

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

      3. difference-of-squares 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in b around inf 54.8%

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

      1. add-sqr-sqrt 47.1%

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

      2. sqrt-unprod 49.7%

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

      3. pow2 49.7%

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

   9. Applied egg-rr 49.7%

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

       1. unpow2 49.7%

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

       2. rem-sqrt-square 49.7%

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

   11. Simplified 49.7%

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

       1. pow1 49.7%

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

       2. add-sqr-sqrt 47.1%

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

       3. fabs-sqr 47.1%

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

       4. add-sqr-sqrt 54.8%

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

       5. 2-sin 54.8%

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

       6. div-inv 52.2%

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

       7. metadata-eval 52.2%

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

   13. Applied egg-rr 52.2%

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

       1. unpow1 52.2%

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

       2. associate-*l* 62.3%

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

       3. associate-*r* 67.5%

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

       4. *-commutative 67.5%

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

       5. associate-*r* 64.9%

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

       6. count-2 64.9%

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

       7. associate-*r* 64.9%

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

       8. *-commutative 64.9%

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

       9. associate-*r* 67.5%

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

       10. *-commutative 67.5%

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

       11. distribute-lft-out 67.5%

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

       12. metadata-eval 67.5%

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

       13. *-commutative 67.5%

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

       14. *-commutative 67.5%

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

   15. Simplified 67.5%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{+155}:\\ \;\;\;\;angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \end{array} \]

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
  (*
   2.0
   (* (* (sin (* PI (* angle_m 0.005555555555555556))) (+ 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 * (2.0 * ((sin((((double) M_PI) * (angle_m * 0.005555555555555556))) * (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 * (2.0 * ((Math.sin((Math.PI * (angle_m * 0.005555555555555556))) * (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 * (2.0 * ((math.sin((math.pi * (angle_m * 0.005555555555555556))) * (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(2.0 * Float64(Float64(sin(Float64(pi * Float64(angle_m * 0.005555555555555556))) * 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 * (2.0 * ((sin((pi * (angle_m * 0.005555555555555556))) * (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[(2.0 * N[(N[(N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. associate-*l* 53.3%

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

   2. *-commutative 53.3%

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

   3. associate-*l* 53.3%

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

3. Simplified 53.3%

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

   1. unpow2 53.3%

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

   2. unpow2 53.3%

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

   3. difference-of-squares 56.5%

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

6. Applied egg-rr 56.5%

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

7. Taylor expanded in angle around inf 56.9%

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

   1. associate-*r* 56.9%

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

   2. *-commutative 56.9%

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

   3. *-commutative 56.9%

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

   4. associate-*r* 55.9%

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

   5. associate-*r* 63.6%

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

   6. *-commutative 63.6%

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

   7. *-commutative 63.6%

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

   8. associate-*r* 62.4%

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

9. Simplified 62.4%

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

10. Taylor expanded in angle around 0 63.9%

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

Alternative 16: 56.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+149}:\\ \;\;\;\;angle\_m \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 2.9e+149)
    (* angle_m (* (* (+ a b) (- b a)) (* PI 0.011111111111111112)))
    (* 0.011111111111111112 (* (* angle_m b) (* PI (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.9e+149) {
		tmp = angle_m * (((a + b) * (b - a)) * (((double) M_PI) * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * b) * (((double) M_PI) * (b - a)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.9e+149) {
		tmp = angle_m * (((a + b) * (b - a)) * (Math.PI * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * b) * (Math.PI * (b - a)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 2.9e+149:
		tmp = angle_m * (((a + b) * (b - a)) * (math.pi * 0.011111111111111112))
	else:
		tmp = 0.011111111111111112 * ((angle_m * b) * (math.pi * (b - a)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 2.9e+149)
		tmp = Float64(angle_m * Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(pi * 0.011111111111111112)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * b) * Float64(pi * Float64(b - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.9e+149)
		tmp = angle_m * (((a + b) * (b - a)) * (pi * 0.011111111111111112));
	else
		tmp = 0.011111111111111112 * ((angle_m * b) * (pi * (b - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 2.9e+149], N[(angle$95$m * N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * b), $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 2.9000000000000002e149

   1. Initial program 56.7%

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

      1. associate-*l* 56.7%

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

      2. *-commutative 56.7%

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

      3. associate-*l* 56.7%

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

   3. Simplified 56.7%

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

      1. unpow2 56.7%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      2. unpow2 56.7%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      3. difference-of-squares 57.7%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   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. Taylor expanded in angle around 0 36.4%

      \[\leadsto \color{blue}{angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. fma-define 36.4%

         \[\leadsto angle \cdot \color{blue}{\mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]

      2. +-commutative 36.4%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      3. *-commutative 36.4%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}, 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      4. +-commutative 36.4%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      5. distribute-rgt-out 36.4%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right), 2 \cdot \left({angle}^{2} \cdot \color{blue}{\left(\left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \left(-8.573388203017833 \cdot 10^{-8} + -2.8577960676726107 \cdot 10^{-8}\right)\right)}\right)\right) \]

   9. Simplified 36.4%

      \[\leadsto \color{blue}{angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(\left({\pi}^{3} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)} \]

   10. Taylor expanded in angle around 0 55.3%

       \[\leadsto angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 55.4%

          \[\leadsto angle \cdot \color{blue}{\left(\left(0.011111111111111112 \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \]

   12. Simplified 55.4%

       \[\leadsto angle \cdot \color{blue}{\left(\left(0.011111111111111112 \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \]

   if 2.9000000000000002e149 < b

   1. Initial program 35.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 35.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 35.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 35.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 35.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 35.1%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      2. unpow2 35.1%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      3. difference-of-squares 49.8%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   6. Applied egg-rr 49.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in b around inf 52.2%

      \[\leadsto \left(\color{blue}{b} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   8. Taylor expanded in angle around 0 52.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 68.7%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]

   10. Simplified 68.7%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+149}:\\ \;\;\;\;angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.8% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+117}:\\ \;\;\;\;angle\_m \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 2.5e+117)
    (* angle_m (* 0.011111111111111112 (* PI (* (+ a b) (- b a)))))
    (* 0.011111111111111112 (* (* angle_m b) (* PI (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.5e+117) {
		tmp = angle_m * (0.011111111111111112 * (((double) M_PI) * ((a + b) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * b) * (((double) M_PI) * (b - a)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.5e+117) {
		tmp = angle_m * (0.011111111111111112 * (Math.PI * ((a + b) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * b) * (Math.PI * (b - a)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 2.5e+117:
		tmp = angle_m * (0.011111111111111112 * (math.pi * ((a + b) * (b - a))))
	else:
		tmp = 0.011111111111111112 * ((angle_m * b) * (math.pi * (b - a)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 2.5e+117)
		tmp = Float64(angle_m * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(a + b) * Float64(b - a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * b) * Float64(pi * Float64(b - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.5e+117)
		tmp = angle_m * (0.011111111111111112 * (pi * ((a + b) * (b - a))));
	else
		tmp = 0.011111111111111112 * ((angle_m * b) * (pi * (b - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 2.5e+117], N[(angle$95$m * N[(0.011111111111111112 * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * b), $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999992e117

   1. Initial program 57.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 57.5%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 57.5%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 57.5%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 57.5%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      2. unpow2 57.5%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      3. difference-of-squares 58.6%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   6. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 36.9%

      \[\leadsto \color{blue}{angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. fma-define 36.9%

         \[\leadsto angle \cdot \color{blue}{\mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]

      2. +-commutative 36.9%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      3. *-commutative 36.9%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}, 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      4. +-commutative 36.9%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(a + b\right)}\right), 2 \cdot \left({angle}^{2} \cdot \left(-8.573388203017833 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + -2.8577960676726107 \cdot 10^{-8} \cdot \left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \]

      5. distribute-rgt-out 36.9%

         \[\leadsto angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right), 2 \cdot \left({angle}^{2} \cdot \color{blue}{\left(\left({\pi}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \left(-8.573388203017833 \cdot 10^{-8} + -2.8577960676726107 \cdot 10^{-8}\right)\right)}\right)\right) \]

   9. Simplified 36.9%

      \[\leadsto \color{blue}{angle \cdot \mathsf{fma}\left(0.011111111111111112, \pi \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right), 2 \cdot \left({angle}^{2} \cdot \left(\left({\pi}^{3} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)} \]

   10. Taylor expanded in angle around 0 56.2%

       \[\leadsto angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   if 2.49999999999999992e117 < b

   1. Initial program 36.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 36.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 36.7%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 36.7%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 36.7%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      2. unpow2 36.7%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      3. difference-of-squares 48.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   6. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in b around inf 50.2%

      \[\leadsto \left(\color{blue}{b} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   8. Taylor expanded in angle around 0 49.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 62.4%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]

   10. Simplified 62.4%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 55.8% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+117}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 4.1e+117)
    (* 0.011111111111111112 (* angle_m (* PI (* (+ a b) (- b a)))))
    (* 0.011111111111111112 (* (* angle_m b) (* PI (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.1e+117) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((a + b) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * b) * (((double) M_PI) * (b - a)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.1e+117) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((a + b) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * b) * (Math.PI * (b - a)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 4.1e+117:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((a + b) * (b - a))))
	else:
		tmp = 0.011111111111111112 * ((angle_m * b) * (math.pi * (b - a)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 4.1e+117)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a + b) * Float64(b - a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * b) * Float64(pi * Float64(b - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 4.1e+117)
		tmp = 0.011111111111111112 * (angle_m * (pi * ((a + b) * (b - a))));
	else
		tmp = 0.011111111111111112 * ((angle_m * b) * (pi * (b - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 4.1e+117], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * b), $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 4.0999999999999999e117

   1. Initial program 57.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0 55.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 57.5%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      2. unpow2 57.5%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      3. difference-of-squares 58.6%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   5. Applied egg-rr 55.8%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   if 4.0999999999999999e117 < b

   1. Initial program 36.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 36.7%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 36.7%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 36.7%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 36.7%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      2. unpow2 36.7%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

      3. difference-of-squares 48.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   6. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in b around inf 50.2%

      \[\leadsto \left(\color{blue}{b} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   8. Taylor expanded in angle around 0 49.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 62.4%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]

   10. Simplified 62.4%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+117}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.3% accurate, 38.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(angle\_m \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* angle_m b) (* PI (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * b) * (((double) M_PI) * (b - a))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * b) * (Math.PI * (b - a))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * ((angle_m * b) * (math.pi * (b - a))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(angle_m * b) * Float64(pi * Float64(b - a)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((angle_m * b) * (pi * (b - a))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(angle$95$m * b), $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 53.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 53.3%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 53.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 53.3%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 53.3%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   2. unpow2 53.3%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   3. difference-of-squares 56.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

6. Applied egg-rr 56.5%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in b around inf 38.9%

   \[\leadsto \left(\color{blue}{b} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

8. Taylor expanded in angle around 0 39.7%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 42.7%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]

10. Simplified 42.7%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]
11. Add Preprocessing

Alternative 20: 37.3% accurate, 38.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(b \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* b (* PI (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (b * (((double) M_PI) * (b - a)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (b * (Math.PI * (b - a)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (b * (math.pi * (b - a)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(b * Float64(pi * Float64(b - a))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (b * (pi * (b - a)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(b * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 53.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 53.3%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 53.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 53.3%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 53.3%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   2. unpow2 53.3%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   3. difference-of-squares 56.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

6. Applied egg-rr 56.5%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in b around inf 38.9%

   \[\leadsto \left(\color{blue}{b} \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

8. Taylor expanded in angle around 0 39.7%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 21: 13.5% accurate, 419.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot 0 \end{array} \]

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 0.0))

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 * 0.0;
}

Fortran

angle\_m = abs(angle)
angle\_s = copysign(1.0d0, angle)
real(8) function code(angle_s, a, b, angle_m)
    real(8), intent (in) :: angle_s
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = angle_s * 0.0d0
end function

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 * 0.0;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * 0.0

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * 0.0)
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 * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 36.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   2. expm1-undefine 23.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} - 1} \]

4. Applied egg-rr 22.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} - 1} \]

5. Taylor expanded in angle around 0 11.8%

   \[\leadsto \color{blue}{1} - 1 \]
6. Step-by-step derivation

   1. metadata-eval 11.8%

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 11.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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)))))