ab-angle->ABCF B

Percentage Accurate: 54.4% → 67.1%

Time: 18.4s

Alternatives: 24

Speedup: 23.3×

• 33.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left({\left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt{\pi} \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt{\pi}\right)\right)\right) \cdot \cos \left(\frac{angle\_m}{180} \cdot \pi\right)\\ \end{array} \end{array} \]

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) 2e+95)
    (*
     (* (sin (* (cbrt (pow PI 3.0)) (* angle_m 0.011111111111111112))) (+ b a))
     (- b a))
    (if (<= (/ angle_m 180.0) 2e+185)
      (*
       (- b a)
       (*
        (+ b a)
        (sin
         (pow
          (pow (* PI (* angle_m 0.011111111111111112)) 0.3333333333333333)
          3.0))))
      (*
       (*
        (* 2.0 (* (+ b a) (- b a)))
        (sin (* (sqrt PI) (* (* angle_m 0.005555555555555556) (sqrt PI)))))
       (cos (* (/ 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) <= 2e+95) {
		tmp = (sin((cbrt(pow(((double) M_PI), 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else if ((angle_m / 180.0) <= 2e+185) {
		tmp = (b - a) * ((b + a) * sin(pow(pow((((double) M_PI) * (angle_m * 0.011111111111111112)), 0.3333333333333333), 3.0)));
	} else {
		tmp = ((2.0 * ((b + a) * (b - a))) * sin((sqrt(((double) M_PI)) * ((angle_m * 0.005555555555555556) * sqrt(((double) M_PI)))))) * cos(((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) <= 2e+95) {
		tmp = (Math.sin((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else if ((angle_m / 180.0) <= 2e+185) {
		tmp = (b - a) * ((b + a) * Math.sin(Math.pow(Math.pow((Math.PI * (angle_m * 0.011111111111111112)), 0.3333333333333333), 3.0)));
	} else {
		tmp = ((2.0 * ((b + a) * (b - a))) * Math.sin((Math.sqrt(Math.PI) * ((angle_m * 0.005555555555555556) * Math.sqrt(Math.PI))))) * Math.cos(((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) <= 2e+95)
		tmp = Float64(Float64(sin(Float64(cbrt((pi ^ 3.0)) * Float64(angle_m * 0.011111111111111112))) * Float64(b + a)) * Float64(b - a));
	elseif (Float64(angle_m / 180.0) <= 2e+185)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(((Float64(pi * Float64(angle_m * 0.011111111111111112)) ^ 0.3333333333333333) ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))) * sin(Float64(sqrt(pi) * Float64(Float64(angle_m * 0.005555555555555556) * sqrt(pi))))) * cos(Float64(Float64(angle_m / 180.0) * pi)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+95], N[(N[(N[Sin[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+185], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000004e95

   1. Initial program 54.5%

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

      1. associate-*l* 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

   3. Simplified 54.5%

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

      1. associate-*r* 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

      4. expm1-log1p-u 36.7%

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

      5. expm1-undefine 25.5%

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

   6. Applied egg-rr 25.0%

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

      1. expm1-define 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-*l* 36.1%

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

      4. metadata-eval 36.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 36.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-*r* 56.1%

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

      4. metadata-eval 56.1%

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

      5. distribute-lft-out 56.1%

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

      6. associate-*r* 55.9%

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

      7. metadata-eval 55.9%

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

      8. div-inv 56.3%

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

      9. associate-*r* 54.8%

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

      10. metadata-eval 54.8%

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

      11. div-inv 54.5%

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

      12. count-2 54.5%

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

      13. 2-sin 54.5%

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

      14. unpow2 54.5%

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

      15. unpow2 54.5%

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

      16. difference-of-squares 58.4%

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

   10. Applied egg-rr 70.5%

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

       1. add-cbrt-cube 73.4%

          \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. pow3 73.4%

          \[\leadsto \left(\sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 73.4%

       \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 2.00000000000000004e95 < (/.f64 angle #s(literal 180 binary64)) < 2e185

   1. Initial program 15.3%

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

      1. associate-*l* 15.3%

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

      2. *-commutative 15.3%

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

      3. associate-*l* 15.3%

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

   3. Simplified 15.3%

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

      1. associate-*r* 15.3%

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

      2. *-commutative 15.3%

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

      3. associate-*l* 15.3%

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

      4. expm1-log1p-u 8.8%

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

      5. expm1-undefine 7.4%

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

   6. Applied egg-rr 7.4%

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

      1. expm1-define 8.7%

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

      2. associate-*l* 8.7%

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

      3. associate-*l* 8.7%

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

      4. metadata-eval 8.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 8.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 20.0%

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

      2. *-commutative 20.0%

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

      3. associate-*r* 20.3%

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

      4. metadata-eval 20.3%

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

      5. distribute-lft-out 20.3%

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

      6. associate-*r* 19.2%

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

      7. metadata-eval 19.2%

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

      8. div-inv 14.4%

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

      9. associate-*r* 15.3%

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

      10. metadata-eval 15.3%

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

      11. div-inv 15.3%

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

      12. count-2 15.3%

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

      13. 2-sin 15.3%

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

      14. unpow2 15.3%

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

      15. unpow2 15.3%

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

      16. difference-of-squares 20.0%

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

   10. Applied egg-rr 29.5%

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

       1. associate-*r* 25.0%

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

       2. *-commutative 25.0%

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

       3. *-commutative 25.0%

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

       4. add-cube-cbrt 35.0%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 34.2%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 34.2%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 34.2%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 32.3%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 32.3%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]
   13. Step-by-step derivation

       1. pow1/3 38.3%

          \[\leadsto \left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   14. Applied egg-rr 38.3%

       \[\leadsto \left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 2e185 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 33.2%

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

      1. unpow2 33.2%

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

      2. unpow2 33.2%

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

      3. difference-of-squares 37.1%

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

   4. Applied egg-rr 37.1%

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

      1. add-cube-cbrt 37.7%

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

      2. pow3 37.7%

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

      3. div-inv 38.2%

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

      4. metadata-eval 38.2%

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

   6. Applied egg-rr 38.2%

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

      1. rem-cube-cbrt 37.6%

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

      2. metadata-eval 37.6%

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

      3. div-inv 37.1%

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

      4. *-commutative 37.1%

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

      5. add-sqr-sqrt 40.8%

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

      6. associate-*r* 41.7%

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

      7. div-inv 41.8%

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

      8. metadata-eval 41.8%

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

   8. Applied egg-rr 41.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\sqrt{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{\pi}\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left({\left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left|b - a\right|\right) \cdot \left(2 \cdot \left(\cos t\_0 \cdot \sin t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+95)
      (*
       (*
        (sin (* (cbrt (pow PI 3.0)) (* angle_m 0.011111111111111112)))
        (+ b a))
       (- b a))
      (if (<= (/ angle_m 180.0) 2e+277)
        (*
         (- b a)
         (*
          (+ b a)
          (sin
           (pow
            (pow (* PI (* angle_m 0.011111111111111112)) 0.3333333333333333)
            3.0))))
        (* (* (+ b a) (fabs (- b a))) (* 2.0 (* (cos t_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 tmp;
	if ((angle_m / 180.0) <= 2e+95) {
		tmp = (sin((cbrt(pow(((double) M_PI), 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else if ((angle_m / 180.0) <= 2e+277) {
		tmp = (b - a) * ((b + a) * sin(pow(pow((((double) M_PI) * (angle_m * 0.011111111111111112)), 0.3333333333333333), 3.0)));
	} else {
		tmp = ((b + a) * fabs((b - a))) * (2.0 * (cos(t_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 tmp;
	if ((angle_m / 180.0) <= 2e+95) {
		tmp = (Math.sin((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else if ((angle_m / 180.0) <= 2e+277) {
		tmp = (b - a) * ((b + a) * Math.sin(Math.pow(Math.pow((Math.PI * (angle_m * 0.011111111111111112)), 0.3333333333333333), 3.0)));
	} else {
		tmp = ((b + a) * Math.abs((b - a))) * (2.0 * (Math.cos(t_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)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+95)
		tmp = Float64(Float64(sin(Float64(cbrt((pi ^ 3.0)) * Float64(angle_m * 0.011111111111111112))) * Float64(b + a)) * Float64(b - a));
	elseif (Float64(angle_m / 180.0) <= 2e+277)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(((Float64(pi * Float64(angle_m * 0.011111111111111112)) ^ 0.3333333333333333) ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(b + a) * abs(Float64(b - a))) * Float64(2.0 * Float64(cos(t_0) * sin(t_0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+95], N[(N[(N[Sin[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+277], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[Abs[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000004e95

   1. Initial program 54.5%

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

      1. associate-*l* 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

   3. Simplified 54.5%

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

      1. associate-*r* 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

      4. expm1-log1p-u 36.7%

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

      5. expm1-undefine 25.5%

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

   6. Applied egg-rr 25.0%

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

      1. expm1-define 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-*l* 36.1%

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

      4. metadata-eval 36.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 36.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-*r* 56.1%

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

      4. metadata-eval 56.1%

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

      5. distribute-lft-out 56.1%

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

      6. associate-*r* 55.9%

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

      7. metadata-eval 55.9%

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

      8. div-inv 56.3%

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

      9. associate-*r* 54.8%

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

      10. metadata-eval 54.8%

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

      11. div-inv 54.5%

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

      12. count-2 54.5%

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

      13. 2-sin 54.5%

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

      14. unpow2 54.5%

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

      15. unpow2 54.5%

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

      16. difference-of-squares 58.4%

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

   10. Applied egg-rr 70.5%

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

       1. add-cbrt-cube 73.4%

          \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. pow3 73.4%

          \[\leadsto \left(\sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 73.4%

       \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 2.00000000000000004e95 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000001e277

   1. Initial program 21.0%

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

      1. associate-*l* 21.0%

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

      2. *-commutative 21.0%

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

      3. associate-*l* 21.0%

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

   3. Simplified 21.0%

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

      1. associate-*r* 21.0%

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

      2. *-commutative 21.0%

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

      3. associate-*l* 21.0%

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

      4. expm1-log1p-u 11.9%

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

      5. expm1-undefine 11.0%

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

   6. Applied egg-rr 11.0%

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

      1. expm1-define 12.3%

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

      2. associate-*l* 12.3%

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

      3. associate-*l* 12.3%

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

      4. metadata-eval 12.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 12.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 23.8%

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

      2. *-commutative 23.8%

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

      3. associate-*r* 22.7%

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

      4. metadata-eval 22.7%

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

      5. distribute-lft-out 22.7%

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

      6. associate-*r* 23.4%

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

      7. metadata-eval 23.4%

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

      8. div-inv 20.5%

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

      9. associate-*r* 21.4%

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

      10. metadata-eval 21.4%

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

      11. div-inv 21.0%

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

      12. count-2 21.0%

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

      13. 2-sin 21.0%

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

      14. unpow2 21.0%

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

      15. unpow2 21.0%

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

      16. difference-of-squares 25.9%

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

   10. Applied egg-rr 31.1%

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

       1. associate-*r* 25.2%

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

       2. *-commutative 25.2%

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

       3. *-commutative 25.2%

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

       4. add-cube-cbrt 28.3%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 28.6%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 28.6%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 28.6%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 30.2%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 30.2%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]
   13. Step-by-step derivation

       1. pow1/3 33.2%

          \[\leadsto \left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   14. Applied egg-rr 33.2%

       \[\leadsto \left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 2.00000000000000001e277 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 54.1%

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

      1. associate-*l* 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-*l* 54.1%

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

   3. Simplified 54.1%

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

      1. unpow2 54.1%

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

      2. unpow2 54.1%

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

      3. difference-of-squares 54.1%

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

   6. Applied egg-rr 54.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 33.5%

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

      3. pow2 33.5%

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

   8. Applied egg-rr 33.5%

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

      1. unpow2 33.5%

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

      2. rem-sqrt-square 33.5%

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

   10. Simplified 33.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left|b - a\right|\right) \cdot \left(2 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left({t\_0}^{0.3333333333333333}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b - a\right| \cdot \left(\left(b + a\right) \cdot \sin t\_0\right)\\ \end{array} \end{array} \end{array} \]

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 (* PI (* angle_m 0.011111111111111112))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+95)
      (*
       (*
        (sin (* (cbrt (pow PI 3.0)) (* angle_m 0.011111111111111112)))
        (+ b a))
       (- b a))
      (if (<= (/ angle_m 180.0) 2e+277)
        (* (- b a) (* (+ b a) (sin (pow (pow t_0 0.3333333333333333) 3.0))))
        (* (fabs (- b a)) (* (+ b a) (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 = ((double) M_PI) * (angle_m * 0.011111111111111112);
	double tmp;
	if ((angle_m / 180.0) <= 2e+95) {
		tmp = (sin((cbrt(pow(((double) M_PI), 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else if ((angle_m / 180.0) <= 2e+277) {
		tmp = (b - a) * ((b + a) * sin(pow(pow(t_0, 0.3333333333333333), 3.0)));
	} else {
		tmp = fabs((b - a)) * ((b + a) * 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 = Math.PI * (angle_m * 0.011111111111111112);
	double tmp;
	if ((angle_m / 180.0) <= 2e+95) {
		tmp = (Math.sin((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else if ((angle_m / 180.0) <= 2e+277) {
		tmp = (b - a) * ((b + a) * Math.sin(Math.pow(Math.pow(t_0, 0.3333333333333333), 3.0)));
	} else {
		tmp = Math.abs((b - a)) * ((b + a) * 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(pi * Float64(angle_m * 0.011111111111111112))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+95)
		tmp = Float64(Float64(sin(Float64(cbrt((pi ^ 3.0)) * Float64(angle_m * 0.011111111111111112))) * Float64(b + a)) * Float64(b - a));
	elseif (Float64(angle_m / 180.0) <= 2e+277)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(((t_0 ^ 0.3333333333333333) ^ 3.0))));
	else
		tmp = Float64(abs(Float64(b - a)) * Float64(Float64(b + a) * sin(t_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 * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+95], N[(N[(N[Sin[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+277], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$0, 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(b - a), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000004e95

   1. Initial program 54.5%

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

      1. associate-*l* 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

   3. Simplified 54.5%

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

      1. associate-*r* 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

      4. expm1-log1p-u 36.7%

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

      5. expm1-undefine 25.5%

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

   6. Applied egg-rr 25.0%

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

      1. expm1-define 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-*l* 36.1%

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

      4. metadata-eval 36.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 36.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-*r* 56.1%

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

      4. metadata-eval 56.1%

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

      5. distribute-lft-out 56.1%

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

      6. associate-*r* 55.9%

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

      7. metadata-eval 55.9%

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

      8. div-inv 56.3%

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

      9. associate-*r* 54.8%

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

      10. metadata-eval 54.8%

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

      11. div-inv 54.5%

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

      12. count-2 54.5%

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

      13. 2-sin 54.5%

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

      14. unpow2 54.5%

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

      15. unpow2 54.5%

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

      16. difference-of-squares 58.4%

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

   10. Applied egg-rr 70.5%

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

       1. add-cbrt-cube 73.4%

          \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. pow3 73.4%

          \[\leadsto \left(\sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 73.4%

       \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 2.00000000000000004e95 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000001e277

   1. Initial program 21.0%

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

      1. associate-*l* 21.0%

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

      2. *-commutative 21.0%

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

      3. associate-*l* 21.0%

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

   3. Simplified 21.0%

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

      1. associate-*r* 21.0%

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

      2. *-commutative 21.0%

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

      3. associate-*l* 21.0%

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

      4. expm1-log1p-u 11.9%

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

      5. expm1-undefine 11.0%

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

   6. Applied egg-rr 11.0%

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

      1. expm1-define 12.3%

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

      2. associate-*l* 12.3%

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

      3. associate-*l* 12.3%

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

      4. metadata-eval 12.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 12.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 23.8%

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

      2. *-commutative 23.8%

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

      3. associate-*r* 22.7%

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

      4. metadata-eval 22.7%

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

      5. distribute-lft-out 22.7%

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

      6. associate-*r* 23.4%

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

      7. metadata-eval 23.4%

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

      8. div-inv 20.5%

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

      9. associate-*r* 21.4%

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

      10. metadata-eval 21.4%

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

      11. div-inv 21.0%

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

      12. count-2 21.0%

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

      13. 2-sin 21.0%

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

      14. unpow2 21.0%

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

      15. unpow2 21.0%

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

      16. difference-of-squares 25.9%

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

   10. Applied egg-rr 31.1%

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

       1. associate-*r* 25.2%

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

       2. *-commutative 25.2%

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

       3. *-commutative 25.2%

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

       4. add-cube-cbrt 28.3%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 28.6%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 28.6%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 28.6%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 30.2%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 30.2%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]
   13. Step-by-step derivation

       1. pow1/3 33.2%

          \[\leadsto \left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   14. Applied egg-rr 33.2%

       \[\leadsto \left(\sin \left({\color{blue}{\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 2.00000000000000001e277 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 54.1%

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

      1. associate-*l* 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-*l* 54.1%

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

   3. Simplified 54.1%

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

      1. associate-*r* 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-*l* 54.1%

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

      4. expm1-log1p-u 33.3%

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

      5. expm1-undefine 33.3%

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

   6. Applied egg-rr 33.3%

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

      1. expm1-define 33.3%

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

      2. associate-*l* 33.3%

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

      3. associate-*l* 33.3%

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

      4. metadata-eval 33.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 33.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 54.1%

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

      2. *-commutative 54.1%

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

      3. associate-*r* 40.4%

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

      4. metadata-eval 40.4%

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

      5. distribute-lft-out 40.4%

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

      6. associate-*r* 54.2%

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

      7. metadata-eval 54.2%

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

      8. div-inv 54.2%

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

      9. associate-*r* 54.1%

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

      10. metadata-eval 54.1%

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

      11. div-inv 54.1%

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

      12. count-2 54.1%

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

      13. 2-sin 54.1%

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

      14. unpow2 54.1%

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

      15. unpow2 54.1%

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

      16. difference-of-squares 54.1%

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

   10. Applied egg-rr 54.1%

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

       1. add-sqr-sqrt 0.1%

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

       2. sqrt-unprod 33.5%

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

       3. pow2 33.5%

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

   12. Applied egg-rr 33.5%

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

       1. unpow2 33.5%

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

       2. rem-sqrt-square 33.5%

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

   14. Simplified 33.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left({\left({\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{0.3333333333333333}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b - a\right| \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b - a\right| \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e+110)
    (*
     (* (sin (* (cbrt (pow PI 3.0)) (* angle_m 0.011111111111111112))) (+ b a))
     (- b a))
    (*
     (fabs (- b a))
     (* (+ b a) (sin (* PI (* angle_m 0.011111111111111112))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+110) {
		tmp = (sin((cbrt(pow(((double) M_PI), 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else {
		tmp = fabs((b - a)) * ((b + a) * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	}
	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) <= 5e+110) {
		tmp = (Math.sin((Math.cbrt(Math.pow(Math.PI, 3.0)) * (angle_m * 0.011111111111111112))) * (b + a)) * (b - a);
	} else {
		tmp = Math.abs((b - a)) * ((b + a) * Math.sin((Math.PI * (angle_m * 0.011111111111111112))));
	}
	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) <= 5e+110)
		tmp = Float64(Float64(sin(Float64(cbrt((pi ^ 3.0)) * Float64(angle_m * 0.011111111111111112))) * Float64(b + a)) * Float64(b - a));
	else
		tmp = Float64(abs(Float64(b - a)) * Float64(Float64(b + a) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+110], N[(N[(N[Sin[N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(b - a), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999978e110

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. *-commutative 53.6%

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

      3. associate-*l* 53.6%

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

   3. Simplified 53.6%

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

      1. associate-*r* 53.6%

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

      2. *-commutative 53.6%

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

      3. associate-*l* 53.6%

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

      4. expm1-log1p-u 36.1%

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

      5. expm1-undefine 25.1%

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

   6. Applied egg-rr 24.7%

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

      1. expm1-define 35.5%

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

      2. associate-*l* 35.5%

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

      3. associate-*l* 35.5%

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

      4. metadata-eval 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 35.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-*r* 55.2%

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

      4. metadata-eval 55.2%

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

      5. distribute-lft-out 55.2%

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

      6. associate-*r* 54.8%

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

      7. metadata-eval 54.8%

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

      8. div-inv 55.3%

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

      9. associate-*r* 53.9%

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

      10. metadata-eval 53.9%

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

      11. div-inv 53.6%

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

      12. count-2 53.6%

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

      13. 2-sin 53.6%

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

      14. unpow2 53.6%

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

      15. unpow2 53.6%

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

      16. difference-of-squares 57.4%

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

   10. Applied egg-rr 69.2%

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

       1. add-cbrt-cube 72.3%

          \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. pow3 72.3%

          \[\leadsto \left(\sin \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 72.3%

       \[\leadsto \left(\sin \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 4.99999999999999978e110 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.1%

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

      1. associate-*l* 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*l* 27.1%

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

   3. Simplified 27.1%

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

      1. associate-*r* 27.1%

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

      2. *-commutative 27.1%

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

      3. associate-*l* 27.1%

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

      4. expm1-log1p-u 15.5%

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

      5. expm1-undefine 14.7%

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

   6. Applied egg-rr 14.7%

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

      1. expm1-define 15.9%

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

      2. associate-*l* 15.9%

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

      3. associate-*l* 15.9%

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

      4. metadata-eval 15.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 15.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 29.8%

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

      2. *-commutative 29.8%

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

      3. associate-*r* 26.8%

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

      4. metadata-eval 26.8%

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

      5. distribute-lft-out 26.8%

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

      6. associate-*r* 29.8%

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

      7. metadata-eval 29.8%

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

      8. div-inv 27.1%

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

      9. associate-*r* 27.5%

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

      10. metadata-eval 27.5%

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

      11. div-inv 27.1%

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

      12. count-2 27.1%

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

      13. 2-sin 27.1%

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

      14. unpow2 27.1%

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

      15. unpow2 27.1%

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

      16. difference-of-squares 31.8%

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

   10. Applied egg-rr 36.8%

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

       1. add-sqr-sqrt 13.5%

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

       2. sqrt-unprod 27.6%

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

       3. pow2 27.6%

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

   12. Applied egg-rr 28.0%

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

       1. unpow2 27.6%

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

       2. rem-sqrt-square 27.6%

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

   14. Simplified 28.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\left(\sin \left(\sqrt[3]{{\pi}^{3}} \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b - a\right| \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+91}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle\_m \cdot \pi}{180}\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+91)
    (*
     (- b a)
     (* (+ b a) (sin (expm1 (log1p (* PI (* angle_m 0.011111111111111112)))))))
    (*
     (* (* 2.0 (* (+ b a) (- b a))) (sin (* (/ angle_m 180.0) PI)))
     (cos (/ (* angle_m PI) 180.0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+91) {
		tmp = (b - a) * ((b + a) * sin(expm1(log1p((((double) M_PI) * (angle_m * 0.011111111111111112))))));
	} else {
		tmp = ((2.0 * ((b + a) * (b - a))) * sin(((angle_m / 180.0) * ((double) M_PI)))) * cos(((angle_m * ((double) M_PI)) / 180.0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+91) {
		tmp = (b - a) * ((b + a) * Math.sin(Math.expm1(Math.log1p((Math.PI * (angle_m * 0.011111111111111112))))));
	} else {
		tmp = ((2.0 * ((b + a) * (b - a))) * Math.sin(((angle_m / 180.0) * Math.PI))) * Math.cos(((angle_m * Math.PI) / 180.0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+91:
		tmp = (b - a) * ((b + a) * math.sin(math.expm1(math.log1p((math.pi * (angle_m * 0.011111111111111112))))))
	else:
		tmp = ((2.0 * ((b + a) * (b - a))) * math.sin(((angle_m / 180.0) * math.pi))) * math.cos(((angle_m * math.pi) / 180.0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+91)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(expm1(log1p(Float64(pi * Float64(angle_m * 0.011111111111111112)))))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))) * sin(Float64(Float64(angle_m / 180.0) * pi))) * cos(Float64(Float64(angle_m * pi) / 180.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+91], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Exp[N[Log[1 + N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000008e91

   1. Initial program 54.7%

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

      1. associate-*l* 54.7%

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

      2. *-commutative 54.7%

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

      3. associate-*l* 54.7%

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

   3. Simplified 54.7%

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

      1. associate-*r* 54.7%

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

      2. *-commutative 54.7%

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

      3. associate-*l* 54.7%

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

      4. expm1-log1p-u 36.9%

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

      5. expm1-undefine 25.7%

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

   6. Applied egg-rr 25.2%

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

      1. expm1-define 36.3%

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

      2. associate-*l* 36.3%

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

      3. associate-*l* 36.3%

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

      4. metadata-eval 36.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*r* 55.9%

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

      4. metadata-eval 55.9%

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

      5. distribute-lft-out 55.9%

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

      6. associate-*r* 55.7%

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

      7. metadata-eval 55.7%

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

      8. div-inv 56.1%

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

      9. associate-*r* 54.6%

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

      10. metadata-eval 54.6%

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

      11. div-inv 54.7%

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

      12. count-2 54.7%

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

      13. 2-sin 54.7%

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

      14. unpow2 54.7%

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

      15. unpow2 54.7%

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

      16. difference-of-squares 58.7%

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

   10. Applied egg-rr 70.8%

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

       1. expm1-log1p-u 61.2%

          \[\leadsto \left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 61.2%

       \[\leadsto \left(\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   if 1.00000000000000008e91 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.7%

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

      1. unpow2 24.7%

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

      2. unpow2 24.7%

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

      3. difference-of-squares 28.8%

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

   4. Applied egg-rr 28.8%

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

      1. associate-*r/ 32.3%

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

   6. Applied egg-rr 32.3%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+91}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+106}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left|\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle\_m \cdot \pi}{180}\right)\\ \end{array} \end{array} \]

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+106)
    (*
     (- b a)
     (* (+ b a) (fabs (sin (* 0.011111111111111112 (* angle_m PI))))))
    (*
     (* (* 2.0 (* (+ b a) (- b a))) (sin (* (/ angle_m 180.0) PI)))
     (cos (/ (* angle_m PI) 180.0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+106) {
		tmp = (b - a) * ((b + a) * fabs(sin((0.011111111111111112 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = ((2.0 * ((b + a) * (b - a))) * sin(((angle_m / 180.0) * ((double) M_PI)))) * cos(((angle_m * ((double) M_PI)) / 180.0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+106) {
		tmp = (b - a) * ((b + a) * Math.abs(Math.sin((0.011111111111111112 * (angle_m * Math.PI)))));
	} else {
		tmp = ((2.0 * ((b + a) * (b - a))) * Math.sin(((angle_m / 180.0) * Math.PI))) * Math.cos(((angle_m * Math.PI) / 180.0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e+106:
		tmp = (b - a) * ((b + a) * math.fabs(math.sin((0.011111111111111112 * (angle_m * math.pi)))))
	else:
		tmp = ((2.0 * ((b + a) * (b - a))) * math.sin(((angle_m / 180.0) * math.pi))) * math.cos(((angle_m * math.pi) / 180.0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+106)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * abs(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b + a) * Float64(b - a))) * sin(Float64(Float64(angle_m / 180.0) * pi))) * cos(Float64(Float64(angle_m * pi) / 180.0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+106)
		tmp = (b - a) * ((b + a) * abs(sin((0.011111111111111112 * (angle_m * pi)))));
	else
		tmp = ((2.0 * ((b + a) * (b - a))) * sin(((angle_m / 180.0) * pi))) * cos(((angle_m * pi) / 180.0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+106], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Abs[N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000036e106

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. *-commutative 53.6%

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

      3. associate-*l* 53.6%

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

   3. Simplified 53.6%

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

      1. associate-*r* 53.6%

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

      2. *-commutative 53.6%

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

      3. associate-*l* 53.6%

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

      4. expm1-log1p-u 36.1%

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

      5. expm1-undefine 25.1%

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

   6. Applied egg-rr 24.7%

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

      1. expm1-define 35.5%

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

      2. associate-*l* 35.5%

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

      3. associate-*l* 35.5%

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

      4. metadata-eval 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 35.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-*r* 55.2%

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

      4. metadata-eval 55.2%

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

      5. distribute-lft-out 55.2%

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

      6. associate-*r* 54.8%

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

      7. metadata-eval 54.8%

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

      8. div-inv 55.3%

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

      9. associate-*r* 53.9%

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

      10. metadata-eval 53.9%

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

      11. div-inv 53.6%

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

      12. count-2 53.6%

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

      13. 2-sin 53.6%

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

      14. unpow2 53.6%

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

      15. unpow2 53.6%

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

      16. difference-of-squares 57.4%

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

   10. Applied egg-rr 69.2%

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

       1. add-sqr-sqrt 40.1%

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

       2. sqrt-unprod 36.4%

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

       3. pow2 36.4%

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

   12. Applied egg-rr 36.4%

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

       1. associate-*r* 36.5%

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

       2. *-commutative 36.5%

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

       3. *-commutative 36.5%

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

       4. unpow2 36.5%

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

       5. rem-sqrt-square 51.7%

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

   14. Simplified 51.7%

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

   if 4.00000000000000036e106 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.1%

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

      1. unpow2 27.1%

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

      2. unpow2 27.1%

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

      3. difference-of-squares 31.8%

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

   4. Applied egg-rr 31.8%

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

      1. associate-*r/ 33.3%

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

   6. Applied egg-rr 33.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+106}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left|\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+127}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left|\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b - a\right| \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e+127)
    (*
     (- b a)
     (* (+ b a) (fabs (sin (* 0.011111111111111112 (* angle_m PI))))))
    (*
     (fabs (- b a))
     (* (+ b a) (sin (* PI (* angle_m 0.011111111111111112))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+127) {
		tmp = (b - a) * ((b + a) * fabs(sin((0.011111111111111112 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = fabs((b - a)) * ((b + a) * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	}
	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+127) {
		tmp = (b - a) * ((b + a) * Math.abs(Math.sin((0.011111111111111112 * (angle_m * Math.PI)))));
	} else {
		tmp = Math.abs((b - a)) * ((b + a) * Math.sin((Math.PI * (angle_m * 0.011111111111111112))));
	}
	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+127:
		tmp = (b - a) * ((b + a) * math.fabs(math.sin((0.011111111111111112 * (angle_m * math.pi)))))
	else:
		tmp = math.fabs((b - a)) * ((b + a) * math.sin((math.pi * (angle_m * 0.011111111111111112))))
	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+127)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * abs(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))))));
	else
		tmp = Float64(abs(Float64(b - a)) * Float64(Float64(b + a) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	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+127)
		tmp = (b - a) * ((b + a) * abs(sin((0.011111111111111112 * (angle_m * pi)))));
	else
		tmp = abs((b - a)) * ((b + a) * sin((pi * (angle_m * 0.011111111111111112))));
	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+127], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Abs[N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(b - a), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999982e127

   1. Initial program 52.5%

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

      1. associate-*l* 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*l* 52.5%

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

   3. Simplified 52.5%

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

      1. associate-*r* 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*l* 52.5%

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

      4. expm1-log1p-u 35.4%

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

      5. expm1-undefine 24.6%

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

   6. Applied egg-rr 24.1%

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

      1. expm1-define 34.8%

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

      2. associate-*l* 34.8%

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

      3. associate-*l* 34.8%

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

      4. metadata-eval 34.8%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 34.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-*r* 54.6%

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

      4. metadata-eval 54.6%

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

      5. distribute-lft-out 54.6%

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

      6. associate-*r* 54.2%

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

      7. metadata-eval 54.2%

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

      8. div-inv 54.2%

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

      9. associate-*r* 52.8%

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

      10. metadata-eval 52.8%

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

      11. div-inv 52.5%

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

      12. count-2 52.5%

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

      13. 2-sin 52.5%

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

      14. unpow2 52.5%

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

      15. unpow2 52.5%

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

      16. difference-of-squares 56.3%

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

   10. Applied egg-rr 68.8%

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

       1. add-sqr-sqrt 39.7%

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

       2. sqrt-unprod 36.2%

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

       3. pow2 36.2%

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

   12. Applied egg-rr 36.2%

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

       1. associate-*r* 36.2%

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

       2. *-commutative 36.2%

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

       3. *-commutative 36.2%

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

       4. unpow2 36.2%

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

       5. rem-sqrt-square 51.1%

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

   14. Simplified 51.1%

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

   if 3.99999999999999982e127 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 29.5%

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

      1. associate-*l* 29.5%

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

      2. *-commutative 29.5%

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

      3. associate-*l* 29.5%

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

   3. Simplified 29.5%

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

      1. associate-*r* 29.5%

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

      2. *-commutative 29.5%

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

      3. associate-*l* 29.5%

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

      4. expm1-log1p-u 16.9%

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

      5. expm1-undefine 16.6%

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

   6. Applied egg-rr 16.5%

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

      1. expm1-define 17.3%

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

      2. associate-*l* 17.3%

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

      3. associate-*l* 17.3%

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

      4. metadata-eval 17.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 17.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 30.0%

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

      2. *-commutative 30.0%

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

      3. associate-*r* 26.6%

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

      4. metadata-eval 26.6%

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

      5. distribute-lft-out 26.6%

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

      6. associate-*r* 30.0%

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

      7. metadata-eval 30.0%

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

      8. div-inv 29.6%

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

      9. associate-*r* 30.0%

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

      10. metadata-eval 30.0%

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

      11. div-inv 29.5%

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

      12. count-2 29.5%

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

      13. 2-sin 29.5%

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

      14. unpow2 29.5%

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

      15. unpow2 29.5%

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

      16. difference-of-squares 34.8%

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

   10. Applied egg-rr 35.2%

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

       1. add-sqr-sqrt 14.8%

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

       2. sqrt-unprod 28.0%

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

       3. pow2 28.0%

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

   12. Applied egg-rr 28.5%

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

       1. unpow2 28.0%

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

       2. rem-sqrt-square 28.0%

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

   14. Simplified 28.5%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+127}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left|\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b - a\right| \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+87}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left|\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(b \cdot \left(\sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \frac{b + a}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+87)
    (*
     (- b a)
     (* (+ b a) (fabs (sin (* 0.011111111111111112 (* angle_m PI))))))
    (*
     (- b a)
     (* b (* (sin (* angle_m (* PI 0.011111111111111112))) (/ (+ b a) b)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+87) {
		tmp = (b - a) * ((b + a) * fabs(sin((0.011111111111111112 * (angle_m * ((double) M_PI))))));
	} else {
		tmp = (b - a) * (b * (sin((angle_m * (((double) M_PI) * 0.011111111111111112))) * ((b + a) / b)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+87) {
		tmp = (b - a) * ((b + a) * Math.abs(Math.sin((0.011111111111111112 * (angle_m * Math.PI)))));
	} else {
		tmp = (b - a) * (b * (Math.sin((angle_m * (Math.PI * 0.011111111111111112))) * ((b + a) / b)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+87:
		tmp = (b - a) * ((b + a) * math.fabs(math.sin((0.011111111111111112 * (angle_m * math.pi)))))
	else:
		tmp = (b - a) * (b * (math.sin((angle_m * (math.pi * 0.011111111111111112))) * ((b + a) / b)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+87)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * abs(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))))));
	else
		tmp = Float64(Float64(b - a) * Float64(b * Float64(sin(Float64(angle_m * Float64(pi * 0.011111111111111112))) * Float64(Float64(b + a) / b))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+87)
		tmp = (b - a) * ((b + a) * abs(sin((0.011111111111111112 * (angle_m * pi)))));
	else
		tmp = (b - a) * (b * (sin((angle_m * (pi * 0.011111111111111112))) * ((b + a) / b)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+87], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Abs[N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(b * N[(N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999996e86

   1. Initial program 54.9%

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

      1. associate-*l* 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-*l* 54.9%

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

   3. Simplified 54.9%

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

      1. associate-*r* 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-*l* 54.9%

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

      4. expm1-log1p-u 36.9%

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

      5. expm1-undefine 25.7%

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

   6. Applied egg-rr 25.3%

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

      1. expm1-define 36.5%

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

      2. associate-*l* 36.5%

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

      3. associate-*l* 36.5%

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

      4. metadata-eval 36.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 36.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-*r* 56.1%

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

      4. metadata-eval 56.1%

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

      5. distribute-lft-out 56.1%

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

      6. associate-*r* 55.9%

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

      7. metadata-eval 55.9%

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

      8. div-inv 56.3%

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

      9. associate-*r* 54.9%

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

      10. metadata-eval 54.9%

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

      11. div-inv 54.9%

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

      12. count-2 54.9%

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

      13. 2-sin 54.9%

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

      14. unpow2 54.9%

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

      15. unpow2 54.9%

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

      16. difference-of-squares 58.9%

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

   10. Applied egg-rr 71.2%

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

       1. add-sqr-sqrt 41.2%

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

       2. sqrt-unprod 36.7%

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

       3. pow2 36.7%

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

   12. Applied egg-rr 36.7%

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

       1. associate-*r* 36.7%

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

       2. *-commutative 36.7%

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

       3. *-commutative 36.7%

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

       4. unpow2 36.7%

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

       5. rem-sqrt-square 52.3%

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

   14. Simplified 52.3%

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

   if 9.9999999999999996e86 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.7%

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

      1. associate-*l* 24.7%

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

      2. *-commutative 24.7%

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

      3. associate-*l* 24.7%

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

   3. Simplified 24.7%

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

      1. associate-*r* 24.7%

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

      2. *-commutative 24.7%

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

      3. associate-*l* 24.7%

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

      4. expm1-log1p-u 14.5%

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

      5. expm1-undefine 13.8%

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

   6. Applied egg-rr 13.3%

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

      1. expm1-define 14.3%

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

      2. associate-*l* 14.3%

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

      3. associate-*l* 14.3%

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

      4. metadata-eval 14.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 14.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 26.6%

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

      2. *-commutative 26.6%

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

      3. associate-*r* 26.5%

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

      4. metadata-eval 26.5%

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

      5. distribute-lft-out 26.5%

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

      6. associate-*r* 28.2%

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

      7. metadata-eval 28.2%

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

      8. div-inv 26.4%

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

      9. associate-*r* 26.6%

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

      10. metadata-eval 26.6%

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

      11. div-inv 24.7%

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

      12. count-2 24.7%

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

      13. 2-sin 24.7%

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

      14. unpow2 24.7%

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

      15. unpow2 24.7%

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

      16. difference-of-squares 28.8%

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

   10. Applied egg-rr 32.7%

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

   11. Taylor expanded in b around inf 28.2%

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

       1. *-commutative 28.2%

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

       2. *-commutative 28.2%

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

       3. associate-*r* 31.1%

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

       4. *-commutative 31.1%

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

       5. associate-/l* 31.1%

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

       6. *-commutative 31.1%

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

       7. *-commutative 31.1%

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

       8. associate-*r* 32.3%

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

       9. *-commutative 32.3%

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

   13. Simplified 32.3%

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

   14. Taylor expanded in b around 0 28.2%

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

       1. distribute-rgt-out 28.2%

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

       2. associate-*r* 32.3%

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

       3. *-commutative 32.3%

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

       4. *-commutative 32.3%

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

       5. associate-/l* 32.3%

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

       6. associate-*r* 28.2%

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

       7. *-commutative 28.2%

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

       8. associate-*r* 30.9%

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

   16. Simplified 30.9%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+87}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left|\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(b \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \frac{b + a}{b}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-164}:\\ \;\;\;\;\left(b - a\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle\_m \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 a 2.0) 2e-164)
    (* (- b a) (* b (sin (* PI (* angle_m 0.011111111111111112)))))
    (* (- b a) (* angle_m (* (+ b a) (* PI 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(a, 2.0) <= 2e-164) {
		tmp = (b - a) * (b * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	} else {
		tmp = (b - a) * (angle_m * ((b + a) * (((double) M_PI) * 0.011111111111111112)));
	}
	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(a, 2.0) <= 2e-164) {
		tmp = (b - a) * (b * Math.sin((Math.PI * (angle_m * 0.011111111111111112))));
	} else {
		tmp = (b - a) * (angle_m * ((b + a) * (Math.PI * 0.011111111111111112)));
	}
	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(a, 2.0) <= 2e-164:
		tmp = (b - a) * (b * math.sin((math.pi * (angle_m * 0.011111111111111112))))
	else:
		tmp = (b - a) * (angle_m * ((b + a) * (math.pi * 0.011111111111111112)))
	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 ((a ^ 2.0) <= 2e-164)
		tmp = Float64(Float64(b - a) * Float64(b * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b - a) * Float64(angle_m * Float64(Float64(b + a) * Float64(pi * 0.011111111111111112))));
	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 ((a ^ 2.0) <= 2e-164)
		tmp = (b - a) * (b * sin((pi * (angle_m * 0.011111111111111112))));
	else
		tmp = (b - a) * (angle_m * ((b + a) * (pi * 0.011111111111111112)));
	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[a, 2.0], $MachinePrecision], 2e-164], N[(N[(b - a), $MachinePrecision] * N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(b + a), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1.99999999999999992e-164

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. associate-*r* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

      4. expm1-log1p-u 44.5%

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

      5. expm1-undefine 37.6%

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

   6. Applied egg-rr 37.3%

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

      1. expm1-define 44.3%

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

      2. associate-*l* 44.3%

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

      3. associate-*l* 44.3%

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

      4. metadata-eval 44.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*r* 56.3%

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

      4. metadata-eval 56.3%

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

      5. distribute-lft-out 56.3%

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

      6. associate-*r* 57.5%

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

      7. metadata-eval 57.5%

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

      8. div-inv 56.5%

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

      9. associate-*r* 56.3%

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

      10. metadata-eval 56.3%

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

      11. div-inv 56.5%

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

      12. count-2 56.5%

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

      13. 2-sin 56.5%

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

      14. unpow2 56.5%

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

      15. unpow2 56.5%

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

      16. difference-of-squares 56.5%

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

   10. Applied egg-rr 65.4%

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

   11. Taylor expanded in b around inf 65.0%

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

   if 1.99999999999999992e-164 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 45.0%

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

      1. associate-*l* 45.0%

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

      2. *-commutative 45.0%

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

      3. associate-*l* 45.0%

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

   3. Simplified 45.0%

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

      1. associate-*r* 45.0%

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

      2. *-commutative 45.0%

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

      3. associate-*l* 45.0%

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

      4. expm1-log1p-u 26.1%

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

      5. expm1-undefine 15.6%

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

   6. Applied egg-rr 15.1%

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

      1. expm1-define 25.6%

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

      2. associate-*l* 25.6%

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

      3. associate-*l* 25.6%

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

      4. metadata-eval 25.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 43.8%

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

      2. *-commutative 43.8%

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

      3. associate-*r* 47.2%

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

      4. metadata-eval 47.2%

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

      5. distribute-lft-out 47.2%

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

      6. associate-*r* 46.8%

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

      7. metadata-eval 46.8%

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

      8. div-inv 47.3%

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

      9. associate-*r* 45.7%

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

      10. metadata-eval 45.7%

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

      11. div-inv 45.0%

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

      12. count-2 45.0%

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

      13. 2-sin 45.0%

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

      14. unpow2 45.0%

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

      15. unpow2 45.0%

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

      16. difference-of-squares 51.2%

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

   10. Applied egg-rr 62.9%

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

       1. associate-*r* 65.0%

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

       2. *-commutative 65.0%

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

       3. *-commutative 65.0%

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

       4. add-cube-cbrt 67.4%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 66.4%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 66.4%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 66.4%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 66.7%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 66.7%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   13. Taylor expanded in angle around 0 61.7%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{0.011111111111111112}\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)} \cdot \left(b - a\right) \]
   14. Step-by-step derivation

       1. rem-cube-cbrt 62.3%

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

       2. associate-*r* 62.4%

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

   15. Simplified 62.4%

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

4. Final simplification 63.3%

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

Alternative 10: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-164}:\\ \;\;\;\;\left(b - a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle\_m \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 a 2.0) 2e-164)
    (* (- b a) (* b (sin (* 0.011111111111111112 (* angle_m PI)))))
    (* (- b a) (* angle_m (* (+ b a) (* PI 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(a, 2.0) <= 2e-164) {
		tmp = (b - a) * (b * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else {
		tmp = (b - a) * (angle_m * ((b + a) * (((double) M_PI) * 0.011111111111111112)));
	}
	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(a, 2.0) <= 2e-164) {
		tmp = (b - a) * (b * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	} else {
		tmp = (b - a) * (angle_m * ((b + a) * (Math.PI * 0.011111111111111112)));
	}
	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(a, 2.0) <= 2e-164:
		tmp = (b - a) * (b * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	else:
		tmp = (b - a) * (angle_m * ((b + a) * (math.pi * 0.011111111111111112)))
	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 ((a ^ 2.0) <= 2e-164)
		tmp = Float64(Float64(b - a) * Float64(b * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	else
		tmp = Float64(Float64(b - a) * Float64(angle_m * Float64(Float64(b + a) * Float64(pi * 0.011111111111111112))));
	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 ((a ^ 2.0) <= 2e-164)
		tmp = (b - a) * (b * sin((0.011111111111111112 * (angle_m * pi))));
	else
		tmp = (b - a) * (angle_m * ((b + a) * (pi * 0.011111111111111112)));
	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[a, 2.0], $MachinePrecision], 2e-164], N[(N[(b - a), $MachinePrecision] * N[(b * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(b + a), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1.99999999999999992e-164

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. associate-*r* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

      4. expm1-log1p-u 44.5%

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

      5. expm1-undefine 37.6%

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

   6. Applied egg-rr 37.3%

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

      1. expm1-define 44.3%

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

      2. associate-*l* 44.3%

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

      3. associate-*l* 44.3%

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

      4. metadata-eval 44.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*r* 56.3%

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

      4. metadata-eval 56.3%

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

      5. distribute-lft-out 56.3%

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

      6. associate-*r* 57.5%

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

      7. metadata-eval 57.5%

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

      8. div-inv 56.5%

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

      9. associate-*r* 56.3%

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

      10. metadata-eval 56.3%

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

      11. div-inv 56.5%

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

      12. count-2 56.5%

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

      13. 2-sin 56.5%

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

      14. unpow2 56.5%

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

      15. unpow2 56.5%

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

      16. difference-of-squares 56.5%

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

   10. Applied egg-rr 65.4%

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

   11. Taylor expanded in b around inf 65.0%

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

       1. *-commutative 65.0%

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

   13. Simplified 65.0%

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

   if 1.99999999999999992e-164 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 45.0%

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

      1. associate-*l* 45.0%

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

      2. *-commutative 45.0%

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

      3. associate-*l* 45.0%

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

   3. Simplified 45.0%

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

      1. associate-*r* 45.0%

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

      2. *-commutative 45.0%

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

      3. associate-*l* 45.0%

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

      4. expm1-log1p-u 26.1%

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

      5. expm1-undefine 15.6%

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

   6. Applied egg-rr 15.1%

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

      1. expm1-define 25.6%

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

      2. associate-*l* 25.6%

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

      3. associate-*l* 25.6%

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

      4. metadata-eval 25.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 43.8%

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

      2. *-commutative 43.8%

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

      3. associate-*r* 47.2%

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

      4. metadata-eval 47.2%

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

      5. distribute-lft-out 47.2%

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

      6. associate-*r* 46.8%

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

      7. metadata-eval 46.8%

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

      8. div-inv 47.3%

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

      9. associate-*r* 45.7%

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

      10. metadata-eval 45.7%

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

      11. div-inv 45.0%

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

      12. count-2 45.0%

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

      13. 2-sin 45.0%

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

      14. unpow2 45.0%

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

      15. unpow2 45.0%

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

      16. difference-of-squares 51.2%

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

   10. Applied egg-rr 62.9%

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

       1. associate-*r* 65.0%

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

       2. *-commutative 65.0%

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

       3. *-commutative 65.0%

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

       4. add-cube-cbrt 67.4%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 66.4%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 66.4%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 66.4%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 66.7%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 66.7%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   13. Taylor expanded in angle around 0 61.7%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{0.011111111111111112}\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)} \cdot \left(b - a\right) \]
   14. Step-by-step derivation

       1. rem-cube-cbrt 62.3%

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

       2. associate-*r* 62.4%

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

   15. Simplified 62.4%

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

4. Final simplification 63.3%

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

Alternative 11: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-216}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle\_m \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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) 5e-216)
    (* (- b a) (* a (sin (* angle_m (* PI 0.011111111111111112)))))
    (* (- b a) (* angle_m (* (+ b a) (* PI 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(b, 2.0) <= 5e-216) {
		tmp = (b - a) * (a * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (b - a) * (angle_m * ((b + a) * (((double) M_PI) * 0.011111111111111112)));
	}
	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) <= 5e-216) {
		tmp = (b - a) * (a * Math.sin((angle_m * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = (b - a) * (angle_m * ((b + a) * (Math.PI * 0.011111111111111112)));
	}
	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) <= 5e-216:
		tmp = (b - a) * (a * math.sin((angle_m * (math.pi * 0.011111111111111112))))
	else:
		tmp = (b - a) * (angle_m * ((b + a) * (math.pi * 0.011111111111111112)))
	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) <= 5e-216)
		tmp = Float64(Float64(b - a) * Float64(a * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b - a) * Float64(angle_m * Float64(Float64(b + a) * Float64(pi * 0.011111111111111112))));
	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) <= 5e-216)
		tmp = (b - a) * (a * sin((angle_m * (pi * 0.011111111111111112))));
	else
		tmp = (b - a) * (angle_m * ((b + a) * (pi * 0.011111111111111112)));
	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], 5e-216], N[(N[(b - a), $MachinePrecision] * N[(a * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(b + a), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 5.00000000000000021e-216

   1. Initial program 64.4%

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

      1. associate-*l* 64.4%

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

      2. *-commutative 64.4%

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

      3. associate-*l* 64.4%

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

   3. Simplified 64.4%

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

      1. associate-*r* 64.4%

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

      2. *-commutative 64.4%

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

      3. associate-*l* 64.4%

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

      4. expm1-log1p-u 47.2%

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

      5. expm1-undefine 37.5%

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

   6. Applied egg-rr 37.5%

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

      1. expm1-define 47.3%

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

      2. associate-*l* 47.3%

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

      3. associate-*l* 47.3%

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

      4. metadata-eval 47.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 47.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 63.1%

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

      2. *-commutative 63.1%

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

      3. associate-*r* 62.7%

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

      4. metadata-eval 62.7%

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

      5. distribute-lft-out 62.7%

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

      6. associate-*r* 63.0%

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

      7. metadata-eval 63.0%

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

      8. div-inv 64.1%

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

      9. associate-*r* 64.4%

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

      10. metadata-eval 64.4%

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

      11. div-inv 64.4%

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

      12. count-2 64.4%

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

      13. 2-sin 64.4%

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

      14. unpow2 64.4%

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

      15. unpow2 64.4%

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

      16. difference-of-squares 64.4%

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

   10. Applied egg-rr 66.8%

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

   11. Taylor expanded in b around inf 65.0%

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

       1. *-commutative 65.0%

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

       2. *-commutative 65.0%

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

       3. associate-*r* 65.2%

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

       4. *-commutative 65.2%

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

       5. associate-/l* 65.2%

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

       6. *-commutative 65.2%

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

       7. *-commutative 65.2%

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

       8. associate-*r* 65.1%

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

       9. *-commutative 65.1%

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

   13. Simplified 65.1%

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

   14. Taylor expanded in b around 0 66.1%

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

       1. *-commutative 66.1%

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

       2. associate-*r* 67.9%

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

       3. rem-cube-cbrt 66.0%

          \[\leadsto \left(a \cdot \sin \left(angle \cdot \left(\pi \cdot \color{blue}{{\left(\sqrt[3]{0.011111111111111112}\right)}^{3}}\right)\right)\right) \cdot \left(b - a\right) \]

       4. rem-cube-cbrt 67.9%

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

       5. associate-*r* 66.1%

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

       6. metadata-eval 66.1%

          \[\leadsto \left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(--0.011111111111111112\right)}\right)\right) \cdot \left(b - a\right) \]

       7. distribute-rgt-neg-in 66.1%

          \[\leadsto \left(a \cdot \sin \color{blue}{\left(-\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)}\right) \cdot \left(b - a\right) \]

       8. associate-*r* 67.9%

          \[\leadsto \left(a \cdot \sin \left(-\color{blue}{angle \cdot \left(\pi \cdot -0.011111111111111112\right)}\right)\right) \cdot \left(b - a\right) \]

       9. rem-cube-cbrt 66.0%

          \[\leadsto \left(a \cdot \sin \left(-angle \cdot \left(\pi \cdot \color{blue}{{\left(\sqrt[3]{-0.011111111111111112}\right)}^{3}}\right)\right)\right) \cdot \left(b - a\right) \]

       10. mul-1-neg 66.0%

           \[\leadsto \left(a \cdot \sin \color{blue}{\left(-1 \cdot \left(angle \cdot \left(\pi \cdot {\left(\sqrt[3]{-0.011111111111111112}\right)}^{3}\right)\right)\right)}\right) \cdot \left(b - a\right) \]

       11. mul-1-neg 66.0%

           \[\leadsto \left(a \cdot \sin \color{blue}{\left(-angle \cdot \left(\pi \cdot {\left(\sqrt[3]{-0.011111111111111112}\right)}^{3}\right)\right)}\right) \cdot \left(b - a\right) \]

       12. rem-cube-cbrt 67.9%

           \[\leadsto \left(a \cdot \sin \left(-angle \cdot \left(\pi \cdot \color{blue}{-0.011111111111111112}\right)\right)\right) \cdot \left(b - a\right) \]

       13. associate-*r* 66.1%

           \[\leadsto \left(a \cdot \sin \left(-\color{blue}{\left(angle \cdot \pi\right) \cdot -0.011111111111111112}\right)\right) \cdot \left(b - a\right) \]

   16. Simplified 67.9%

       \[\leadsto \color{blue}{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \cdot \left(b - a\right) \]

   if 5.00000000000000021e-216 < (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. associate-*r* 42.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 42.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 42.8%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      4. expm1-log1p-u 26.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      5. expm1-undefine 17.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} - 1} \]

   6. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 26.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)} \]

      2. associate-*l* 26.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot 2\right)\right)}\right)\right) \]

      3. associate-*l* 26.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right)\right)\right) \]

      4. metadata-eval 26.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 42.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

      2. *-commutative 42.1%

         \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. associate-*r* 45.3%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      4. metadata-eval 45.3%

         \[\leadsto \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(0.005555555555555556 + 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. distribute-lft-out 45.3%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. associate-*r* 45.5%

         \[\leadsto \sin \left(\color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. metadata-eval 45.5%

         \[\leadsto \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. div-inv 44.9%

         \[\leadsto \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. associate-*r* 43.2%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. metadata-eval 43.2%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. div-inv 42.8%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      12. count-2 42.8%

          \[\leadsto \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      13. 2-sin 42.8%

          \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      14. unpow2 42.8%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      15. unpow2 42.8%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      16. difference-of-squares 48.4%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. Applied egg-rr 62.5%

       \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 64.7%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. *-commutative 64.7%

          \[\leadsto \left(\sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       3. *-commutative 64.7%

          \[\leadsto \left(\sin \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       4. add-cube-cbrt 67.8%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 66.2%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 66.2%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 66.2%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 66.0%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 66.0%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   13. Taylor expanded in angle around 0 60.6%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{0.011111111111111112}\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)} \cdot \left(b - a\right) \]
   14. Step-by-step derivation

       1. rem-cube-cbrt 61.2%

          \[\leadsto \left(angle \cdot \left(\pi \cdot \left(\color{blue}{0.011111111111111112} \cdot \left(a + b\right)\right)\right)\right) \cdot \left(b - a\right) \]

       2. associate-*r* 61.3%

          \[\leadsto \left(angle \cdot \color{blue}{\left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)}\right) \cdot \left(b - a\right) \]

   15. Simplified 61.3%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(b - a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-216}:\\ \;\;\;\;\left(b - a\right) \cdot \left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle\_m \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \end{array} \end{array} \]

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 1.5e-12)
    (* (- b a) (* angle_m (* (+ b a) (* PI 0.011111111111111112))))
    (* (* (+ b a) (- 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 (angle_m <= 1.5e-12) {
		tmp = (b - a) * (angle_m * ((b + a) * (((double) M_PI) * 0.011111111111111112)));
	} else {
		tmp = ((b + a) * (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 (angle_m <= 1.5e-12) {
		tmp = (b - a) * (angle_m * ((b + a) * (Math.PI * 0.011111111111111112)));
	} else {
		tmp = ((b + a) * (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 angle_m <= 1.5e-12:
		tmp = (b - a) * (angle_m * ((b + a) * (math.pi * 0.011111111111111112)))
	else:
		tmp = ((b + a) * (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 (angle_m <= 1.5e-12)
		tmp = Float64(Float64(b - a) * Float64(angle_m * Float64(Float64(b + a) * Float64(pi * 0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(b + a) * 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 (angle_m <= 1.5e-12)
		tmp = (b - a) * (angle_m * ((b + a) * (pi * 0.011111111111111112)));
	else
		tmp = ((b + a) * (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[angle$95$m, 1.5e-12], N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(b + a), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.5000000000000001e-12

   1. Initial program 56.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 56.4%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 56.4%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 56.4%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 56.4%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 56.4%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 56.4%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      4. expm1-log1p-u 39.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      5. expm1-undefine 27.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} - 1} \]

   6. Applied egg-rr 27.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 39.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)} \]

      2. associate-*l* 39.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot 2\right)\right)}\right)\right) \]

      3. associate-*l* 39.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right)\right)\right) \]

      4. metadata-eval 39.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 55.3%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

      2. *-commutative 55.3%

         \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. associate-*r* 58.2%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      4. metadata-eval 58.2%

         \[\leadsto \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(0.005555555555555556 + 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. distribute-lft-out 58.2%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. associate-*r* 58.1%

         \[\leadsto \sin \left(\color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. metadata-eval 58.1%

         \[\leadsto \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. div-inv 57.9%

         \[\leadsto \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. associate-*r* 56.4%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. metadata-eval 56.4%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. div-inv 56.4%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      12. count-2 56.4%

          \[\leadsto \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      13. 2-sin 56.4%

          \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      14. unpow2 56.4%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      15. unpow2 56.4%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      16. difference-of-squares 60.7%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 77.6%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. *-commutative 77.6%

          \[\leadsto \left(\sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       3. *-commutative 77.6%

          \[\leadsto \left(\sin \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       4. add-cube-cbrt 76.9%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 75.8%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 75.8%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 75.8%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 75.8%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 75.8%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   13. Taylor expanded in angle around 0 70.7%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{0.011111111111111112}\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)} \cdot \left(b - a\right) \]
   14. Step-by-step derivation

       1. rem-cube-cbrt 71.3%

          \[\leadsto \left(angle \cdot \left(\pi \cdot \left(\color{blue}{0.011111111111111112} \cdot \left(a + b\right)\right)\right)\right) \cdot \left(b - a\right) \]

       2. associate-*r* 71.5%

          \[\leadsto \left(angle \cdot \color{blue}{\left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)}\right) \cdot \left(b - a\right) \]

   15. Simplified 71.5%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(b - a\right) \]

   if 1.5000000000000001e-12 < angle

   1. Initial program 28.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 28.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 28.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 28.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 28.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 28.1%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 28.1%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 31.2%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. 2-sin 31.2%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. count-2 31.2%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)} \]

      3. div-inv 32.6%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)} + \pi \cdot \frac{angle}{180}\right) \]

      4. metadata-eval 32.6%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right) + \pi \cdot \frac{angle}{180}\right) \]

      5. associate-*r* 32.4%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\color{blue}{\left(\pi \cdot angle\right) \cdot 0.005555555555555556} + \pi \cdot \frac{angle}{180}\right) \]

      6. div-inv 32.2%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) \]

      7. metadata-eval 32.2%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right) \]

      8. associate-*r* 29.5%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \color{blue}{\left(\pi \cdot angle\right) \cdot 0.005555555555555556}\right) \]

      9. distribute-lft-out 29.5%

         \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(0.005555555555555556 + 0.005555555555555556\right)\right)} \]

      10. metadata-eval 29.5%

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{0.011111111111111112}\right) \]

      11. associate-*r* 32.6%

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

      12. log1p-expm1-u 32.6%

          \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]

   8. Applied egg-rr 32.6%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]

   9. Taylor expanded in angle around inf 29.5%

      \[\leadsto \color{blue}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\right) \end{array} \]

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
  (* (- b a) (* (+ b a) (sin (* PI (* angle_m 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b - a) * ((b + a) * sin((((double) M_PI) * (angle_m * 0.011111111111111112)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b - a) * ((b + a) * Math.sin((Math.PI * (angle_m * 0.011111111111111112)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((b - a) * ((b + a) * math.sin((math.pi * (angle_m * 0.011111111111111112)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((b - a) * ((b + a) * sin((pi * (angle_m * 0.011111111111111112)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 49.1%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 49.1%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* 49.1%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 49.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

   4. expm1-log1p-u 32.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   5. expm1-undefine 23.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} - 1} \]

6. Applied egg-rr 23.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-define 32.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)} \]

   2. associate-*l* 32.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot 2\right)\right)}\right)\right) \]

   3. associate-*l* 32.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right)\right)\right) \]

   4. metadata-eval 32.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

8. Simplified 32.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. expm1-log1p-u 48.3%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

   2. *-commutative 48.3%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   3. associate-*r* 50.4%

      \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   4. metadata-eval 50.4%

      \[\leadsto \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(0.005555555555555556 + 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   5. distribute-lft-out 50.4%

      \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   6. associate-*r* 50.6%

      \[\leadsto \sin \left(\color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   7. metadata-eval 50.6%

      \[\leadsto \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   8. div-inv 50.6%

      \[\leadsto \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   9. associate-*r* 49.4%

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   10. metadata-eval 49.4%

       \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   11. div-inv 49.1%

       \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   12. count-2 49.1%

       \[\leadsto \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   13. 2-sin 49.1%

       \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   14. unpow2 49.1%

       \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   15. unpow2 49.1%

       \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   16. difference-of-squares 53.1%

       \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

10. Applied egg-rr 63.8%

    \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \]

11. Final simplification 63.8%

    \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \]
12. Add Preprocessing

Alternative 14: 62.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle\_m \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

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 2.2e+109)
    (* (- b a) (* angle_m (* (+ b a) (* PI 0.011111111111111112))))
    (* (* 0.011111111111111112 (* angle_m 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 (angle_m <= 2.2e+109) {
		tmp = (b - a) * (angle_m * ((b + a) * (((double) M_PI) * 0.011111111111111112)));
	} else {
		tmp = (0.011111111111111112 * (angle_m * ((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 (angle_m <= 2.2e+109) {
		tmp = (b - a) * (angle_m * ((b + a) * (Math.PI * 0.011111111111111112)));
	} else {
		tmp = (0.011111111111111112 * (angle_m * 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 angle_m <= 2.2e+109:
		tmp = (b - a) * (angle_m * ((b + a) * (math.pi * 0.011111111111111112)))
	else:
		tmp = (0.011111111111111112 * (angle_m * 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 (angle_m <= 2.2e+109)
		tmp = Float64(Float64(b - a) * Float64(angle_m * Float64(Float64(b + a) * Float64(pi * 0.011111111111111112))));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * 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 (angle_m <= 2.2e+109)
		tmp = (b - a) * (angle_m * ((b + a) * (pi * 0.011111111111111112)));
	else
		tmp = (0.011111111111111112 * (angle_m * 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[angle$95$m, 2.2e+109], N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(b + a), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.1999999999999999e109

   1. Initial program 53.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 53.6%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 53.6%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 53.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      4. expm1-log1p-u 36.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      5. expm1-undefine 25.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} - 1} \]

   6. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 35.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)} \]

      2. associate-*l* 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot 2\right)\right)}\right)\right) \]

      3. associate-*l* 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right)\right)\right) \]

      4. metadata-eval 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 35.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 52.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

      2. *-commutative 52.0%

         \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. associate-*r* 55.2%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      4. metadata-eval 55.2%

         \[\leadsto \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(0.005555555555555556 + 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. distribute-lft-out 55.2%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. associate-*r* 54.8%

         \[\leadsto \sin \left(\color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. metadata-eval 54.8%

         \[\leadsto \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. div-inv 55.3%

         \[\leadsto \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. associate-*r* 53.9%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. metadata-eval 53.9%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. div-inv 53.6%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      12. count-2 53.6%

          \[\leadsto \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      13. 2-sin 53.6%

          \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      14. unpow2 53.6%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      15. unpow2 53.6%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      16. difference-of-squares 57.4%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. Applied egg-rr 69.2%

       \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 72.4%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. *-commutative 72.4%

          \[\leadsto \left(\sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       3. *-commutative 72.4%

          \[\leadsto \left(\sin \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       4. add-cube-cbrt 73.0%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 72.0%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 72.0%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 72.0%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 71.8%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 71.8%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   13. Taylor expanded in angle around 0 66.4%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{0.011111111111111112}\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)} \cdot \left(b - a\right) \]
   14. Step-by-step derivation

       1. rem-cube-cbrt 67.0%

          \[\leadsto \left(angle \cdot \left(\pi \cdot \left(\color{blue}{0.011111111111111112} \cdot \left(a + b\right)\right)\right)\right) \cdot \left(b - a\right) \]

       2. associate-*r* 67.1%

          \[\leadsto \left(angle \cdot \color{blue}{\left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)}\right) \cdot \left(b - a\right) \]

   15. Simplified 67.1%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(b - a\right) \]

   if 2.1999999999999999e109 < angle

   1. Initial program 27.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 27.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 31.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 27.0%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 22.4%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in a around 0 18.1%

      \[\leadsto \color{blue}{\left(a \cdot b\right)} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;\left(b - a\right) \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

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 2.2e+109)
    (* (- b a) (* 0.011111111111111112 (* angle_m (* PI (+ b a)))))
    (* (* 0.011111111111111112 (* angle_m 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 (angle_m <= 2.2e+109) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (b + a))));
	} else {
		tmp = (0.011111111111111112 * (angle_m * ((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 (angle_m <= 2.2e+109) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (Math.PI * (b + a))));
	} else {
		tmp = (0.011111111111111112 * (angle_m * 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 angle_m <= 2.2e+109:
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (math.pi * (b + a))))
	else:
		tmp = (0.011111111111111112 * (angle_m * 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 (angle_m <= 2.2e+109)
		tmp = Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b + a)))));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * 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 (angle_m <= 2.2e+109)
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (pi * (b + a))));
	else
		tmp = (0.011111111111111112 * (angle_m * 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[angle$95$m, 2.2e+109], N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.1999999999999999e109

   1. Initial program 53.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 53.6%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 53.6%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 53.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      4. expm1-log1p-u 36.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      5. expm1-undefine 25.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} - 1} \]

   6. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 35.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)} \]

      2. associate-*l* 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot 2\right)\right)}\right)\right) \]

      3. associate-*l* 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right)\right)\right) \]

      4. metadata-eval 35.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 35.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 52.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

      2. *-commutative 52.0%

         \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. associate-*r* 55.2%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      4. metadata-eval 55.2%

         \[\leadsto \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(0.005555555555555556 + 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. distribute-lft-out 55.2%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. associate-*r* 54.8%

         \[\leadsto \sin \left(\color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. metadata-eval 54.8%

         \[\leadsto \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. div-inv 55.3%

         \[\leadsto \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. associate-*r* 53.9%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. metadata-eval 53.9%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. div-inv 53.6%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      12. count-2 53.6%

          \[\leadsto \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      13. 2-sin 53.6%

          \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      14. unpow2 53.6%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      15. unpow2 53.6%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      16. difference-of-squares 57.4%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. Applied egg-rr 69.2%

       \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \]

   11. Taylor expanded in angle around 0 67.0%

       \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \cdot \left(b - a\right) \]

   if 2.1999999999999999e109 < angle

   1. Initial program 27.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 27.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 31.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 27.0%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 22.4%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in a around 0 18.1%

      \[\leadsto \color{blue}{\left(a \cdot b\right)} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+148}:\\ \;\;\;\;angle\_m \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2e+148)
    (* angle_m (* PI (* 0.011111111111111112 (* (+ b a) (- b a)))))
    (* 0.011111111111111112 (* a (* angle_m (* 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 (a <= 2e+148) {
		tmp = angle_m * (((double) M_PI) * (0.011111111111111112 * ((b + a) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * (a * (angle_m * (((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 (a <= 2e+148) {
		tmp = angle_m * (Math.PI * (0.011111111111111112 * ((b + a) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * (a * (angle_m * (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 a <= 2e+148:
		tmp = angle_m * (math.pi * (0.011111111111111112 * ((b + a) * (b - a))))
	else:
		tmp = 0.011111111111111112 * (a * (angle_m * (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 (a <= 2e+148)
		tmp = Float64(angle_m * Float64(pi * Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(b - a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(angle_m * 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 (a <= 2e+148)
		tmp = angle_m * (pi * (0.011111111111111112 * ((b + a) * (b - a))));
	else
		tmp = 0.011111111111111112 * (a * (angle_m * (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[a, 2e+148], N[(angle$95$m * N[(Pi * N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(angle$95$m * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.0000000000000001e148

   1. Initial program 50.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 50.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 50.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 50.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 50.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 50.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 50.9%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      4. expm1-log1p-u 35.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

      5. expm1-undefine 24.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} - 1} \]

   6. Applied egg-rr 23.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-define 34.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)} \]

      2. associate-*l* 34.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot 2\right)\right)}\right)\right) \]

      3. associate-*l* 34.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot 2\right)\right)}\right)\right)\right) \]

      4. metadata-eval 34.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.011111111111111112}\right)\right)\right)\right) \]

   8. Simplified 34.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 49.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)} \]

      2. *-commutative 49.9%

         \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. associate-*r* 52.4%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      4. metadata-eval 52.4%

         \[\leadsto \sin \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(0.005555555555555556 + 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. distribute-lft-out 52.4%

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556 + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. associate-*r* 52.7%

         \[\leadsto \sin \left(\color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. metadata-eval 52.7%

         \[\leadsto \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right) + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      8. div-inv 52.6%

         \[\leadsto \sin \left(\pi \cdot \color{blue}{\frac{angle}{180}} + \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      9. associate-*r* 51.3%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \color{blue}{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      10. metadata-eval 51.3%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      11. div-inv 50.9%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      12. count-2 50.9%

          \[\leadsto \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      13. 2-sin 50.9%

          \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      14. unpow2 50.9%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      15. unpow2 50.9%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      16. difference-of-squares 52.8%

          \[\leadsto \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. Applied egg-rr 61.4%

       \[\leadsto \color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 63.5%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       2. *-commutative 63.5%

          \[\leadsto \left(\sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       3. *-commutative 63.5%

          \[\leadsto \left(\sin \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       4. add-cube-cbrt 63.9%

          \[\leadsto \left(\sin \color{blue}{\left(\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)} \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right) \cdot \sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       5. pow3 63.1%

          \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{0.011111111111111112 \cdot \left(angle \cdot \pi\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       6. *-commutative 63.1%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(angle \cdot \pi\right) \cdot 0.011111111111111112}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       7. *-commutative 63.1%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

       8. associate-*r* 63.8%

          \[\leadsto \left(\sin \left({\left(\sqrt[3]{\color{blue}{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}}\right)}^{3}\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   12. Applied egg-rr 63.8%

       \[\leadsto \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.011111111111111112\right)}\right)}^{3}\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right) \]

   13. Taylor expanded in angle around 0 51.0%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left({\left(\sqrt[3]{0.011111111111111112}\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   14. Step-by-step derivation

       1. rem-cube-cbrt 51.4%

          \[\leadsto angle \cdot \left(\pi \cdot \left(\color{blue}{0.011111111111111112} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       2. *-commutative 51.4%

          \[\leadsto angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right)\right) \]

   15. Simplified 51.4%

       \[\leadsto \color{blue}{angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)\right)} \]

   if 2.0000000000000001e148 < a

   1. Initial program 38.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 38.0%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 38.0%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 38.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 38.0%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 38.0%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 55.1%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 43.3%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 35.0%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in angle around 0 55.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+148}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2.25e+148)
    (* 0.011111111111111112 (* angle_m (* PI (* (+ b a) (- b a)))))
    (* 0.011111111111111112 (* a (* angle_m (* 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 (a <= 2.25e+148) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((b + a) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * (a * (angle_m * (((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 (a <= 2.25e+148) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((b + a) * (b - a))));
	} else {
		tmp = 0.011111111111111112 * (a * (angle_m * (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 a <= 2.25e+148:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((b + a) * (b - a))))
	else:
		tmp = 0.011111111111111112 * (a * (angle_m * (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 (a <= 2.25e+148)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b + a) * Float64(b - a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(angle_m * 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 (a <= 2.25e+148)
		tmp = 0.011111111111111112 * (angle_m * (pi * ((b + a) * (b - a))));
	else
		tmp = 0.011111111111111112 * (a * (angle_m * (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[a, 2.25e+148], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(a * N[(angle$95$m * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.24999999999999997e148

   1. Initial program 50.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 50.9%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 50.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 50.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 50.9%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 50.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 52.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 52.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 51.3%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in angle around 0 51.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   if 2.24999999999999997e148 < a

   1. Initial program 38.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 38.0%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 38.0%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 38.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 38.0%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 38.0%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 55.1%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 43.3%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 35.0%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in angle around 0 55.5%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.1% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 0.00058:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 0.00058)
    (* (* 0.011111111111111112 (* angle_m PI)) (* b (- b a)))
    (* a (* (- b a) (* angle_m (* PI 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 0.00058) {
		tmp = (0.011111111111111112 * (angle_m * ((double) M_PI))) * (b * (b - a));
	} else {
		tmp = a * ((b - a) * (angle_m * (((double) M_PI) * 0.011111111111111112)));
	}
	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 (a <= 0.00058) {
		tmp = (0.011111111111111112 * (angle_m * Math.PI)) * (b * (b - a));
	} else {
		tmp = a * ((b - a) * (angle_m * (Math.PI * 0.011111111111111112)));
	}
	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 a <= 0.00058:
		tmp = (0.011111111111111112 * (angle_m * math.pi)) * (b * (b - a))
	else:
		tmp = a * ((b - a) * (angle_m * (math.pi * 0.011111111111111112)))
	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 (a <= 0.00058)
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * pi)) * Float64(b * Float64(b - a)));
	else
		tmp = Float64(a * Float64(Float64(b - a) * Float64(angle_m * Float64(pi * 0.011111111111111112))));
	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 (a <= 0.00058)
		tmp = (0.011111111111111112 * (angle_m * pi)) * (b * (b - a));
	else
		tmp = a * ((b - a) * (angle_m * (pi * 0.011111111111111112)));
	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[a, 0.00058], N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.8e-4

   1. Initial program 50.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 50.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 50.6%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 50.6%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 50.6%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 50.6%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 50.6%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 52.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 52.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 51.3%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around inf 40.7%

      \[\leadsto \left(\color{blue}{b} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   if 5.8e-4 < a

   1. Initial program 45.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 45.4%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 45.4%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 45.4%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 45.4%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 45.4%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 45.4%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 53.7%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 47.6%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 39.6%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
   9. Step-by-step derivation

      1. pow1 39.6%

         \[\leadsto \color{blue}{{\left(\left(a \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{1}} \]

      2. associate-*l* 48.3%

         \[\leadsto {\color{blue}{\left(a \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}^{1} \]

      3. *-commutative 48.3%

         \[\leadsto {\left(a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)\right)}^{1} \]

      4. *-commutative 48.3%

         \[\leadsto {\left(a \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112\right)\right)\right)}^{1} \]

      5. associate-*r* 48.3%

         \[\leadsto {\left(a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}\right)\right)}^{1} \]

   10. Applied egg-rr 48.3%

       \[\leadsto \color{blue}{{\left(a \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 48.3%

          \[\leadsto \color{blue}{a \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

       2. associate-*r* 48.3%

          \[\leadsto a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}\right) \]

       3. *-commutative 48.3%

          \[\leadsto a \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right)\right) \]

       4. associate-*r* 48.4%

          \[\leadsto a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]

   12. Simplified 48.4%

       \[\leadsto \color{blue}{a \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.00058:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.3% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;a \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

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 2.2e+109)
    (* a (* (- b a) (* angle_m (* PI 0.011111111111111112))))
    (* (* 0.011111111111111112 (* angle_m 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 (angle_m <= 2.2e+109) {
		tmp = a * ((b - a) * (angle_m * (((double) M_PI) * 0.011111111111111112)));
	} else {
		tmp = (0.011111111111111112 * (angle_m * ((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 (angle_m <= 2.2e+109) {
		tmp = a * ((b - a) * (angle_m * (Math.PI * 0.011111111111111112)));
	} else {
		tmp = (0.011111111111111112 * (angle_m * 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 angle_m <= 2.2e+109:
		tmp = a * ((b - a) * (angle_m * (math.pi * 0.011111111111111112)))
	else:
		tmp = (0.011111111111111112 * (angle_m * 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 (angle_m <= 2.2e+109)
		tmp = Float64(a * Float64(Float64(b - a) * Float64(angle_m * Float64(pi * 0.011111111111111112))));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * 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 (angle_m <= 2.2e+109)
		tmp = a * ((b - a) * (angle_m * (pi * 0.011111111111111112)));
	else
		tmp = (0.011111111111111112 * (angle_m * 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[angle$95$m, 2.2e+109], N[(a * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.1999999999999999e109

   1. Initial program 53.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 53.6%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 53.6%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 53.6%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 57.4%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 57.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 54.9%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 38.0%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
   9. Step-by-step derivation

      1. pow1 38.0%

         \[\leadsto \color{blue}{{\left(\left(a \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{1}} \]

      2. associate-*l* 42.3%

         \[\leadsto {\color{blue}{\left(a \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}^{1} \]

      3. *-commutative 42.3%

         \[\leadsto {\left(a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)\right)}^{1} \]

      4. *-commutative 42.3%

         \[\leadsto {\left(a \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112\right)\right)\right)}^{1} \]

      5. associate-*r* 42.4%

         \[\leadsto {\left(a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}\right)\right)}^{1} \]

   10. Applied egg-rr 42.4%

       \[\leadsto \color{blue}{{\left(a \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 42.4%

          \[\leadsto \color{blue}{a \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)} \]

       2. associate-*r* 42.3%

          \[\leadsto a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}\right) \]

       3. *-commutative 42.3%

          \[\leadsto a \cdot \left(\left(b - a\right) \cdot \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.011111111111111112\right)\right) \]

       4. associate-*r* 42.4%

          \[\leadsto a \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]

   12. Simplified 42.4%

       \[\leadsto \color{blue}{a \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \]

   if 2.1999999999999999e109 < angle

   1. Initial program 27.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 27.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 31.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 27.0%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 22.4%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in a around 0 18.1%

      \[\leadsto \color{blue}{\left(a \cdot b\right)} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;a \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 40.2% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

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 2.2e+109)
    (* 0.011111111111111112 (* a (* angle_m (* PI (- b a)))))
    (* (* 0.011111111111111112 (* angle_m 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 (angle_m <= 2.2e+109) {
		tmp = 0.011111111111111112 * (a * (angle_m * (((double) M_PI) * (b - a))));
	} else {
		tmp = (0.011111111111111112 * (angle_m * ((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 (angle_m <= 2.2e+109) {
		tmp = 0.011111111111111112 * (a * (angle_m * (Math.PI * (b - a))));
	} else {
		tmp = (0.011111111111111112 * (angle_m * 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 angle_m <= 2.2e+109:
		tmp = 0.011111111111111112 * (a * (angle_m * (math.pi * (b - a))))
	else:
		tmp = (0.011111111111111112 * (angle_m * 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 (angle_m <= 2.2e+109)
		tmp = Float64(0.011111111111111112 * Float64(a * Float64(angle_m * Float64(pi * Float64(b - a)))));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * 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 (angle_m <= 2.2e+109)
		tmp = 0.011111111111111112 * (a * (angle_m * (pi * (b - a))));
	else
		tmp = (0.011111111111111112 * (angle_m * 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[angle$95$m, 2.2e+109], N[(0.011111111111111112 * N[(a * N[(angle$95$m * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.1999999999999999e109

   1. Initial program 53.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 53.6%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 53.6%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 53.6%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 53.6%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 57.4%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 57.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 54.9%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 38.0%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in angle around 0 42.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 2.1999999999999999e109 < angle

   1. Initial program 27.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 27.1%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 27.1%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 27.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 27.1%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 31.8%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 27.0%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 22.4%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in a around 0 18.1%

      \[\leadsto \color{blue}{\left(a \cdot b\right)} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.6% accurate, 27.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{+156}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(-0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

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 3.1e+156)
    (* (* a a) (* (* angle_m PI) (- 0.011111111111111112)))
    (* (* 0.011111111111111112 (* angle_m 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 <= 3.1e+156) {
		tmp = (a * a) * ((angle_m * ((double) M_PI)) * -0.011111111111111112);
	} else {
		tmp = (0.011111111111111112 * (angle_m * ((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 <= 3.1e+156) {
		tmp = (a * a) * ((angle_m * Math.PI) * -0.011111111111111112);
	} else {
		tmp = (0.011111111111111112 * (angle_m * 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 <= 3.1e+156:
		tmp = (a * a) * ((angle_m * math.pi) * -0.011111111111111112)
	else:
		tmp = (0.011111111111111112 * (angle_m * 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 <= 3.1e+156)
		tmp = Float64(Float64(a * a) * Float64(Float64(angle_m * pi) * Float64(-0.011111111111111112)));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * 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 <= 3.1e+156)
		tmp = (a * a) * ((angle_m * pi) * -0.011111111111111112);
	else
		tmp = (0.011111111111111112 * (angle_m * 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, 3.1e+156], N[(N[(a * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * (-0.011111111111111112)), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 3.1000000000000002e156

   1. Initial program 51.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 51.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 51.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 51.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 51.8%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 51.8%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 54.5%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 54.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 52.1%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 38.2%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in b around 0 36.7%

      \[\leadsto \left(a \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 36.7%

          \[\leadsto \left(a \cdot \color{blue}{\left(-a\right)}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   11. Simplified 36.7%

       \[\leadsto \left(a \cdot \color{blue}{\left(-a\right)}\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   if 3.1000000000000002e156 < b

   1. Initial program 32.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 32.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. *-commutative 32.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. associate-*l* 32.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

   3. Simplified 32.8%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow2 32.8%

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. unpow2 32.8%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. difference-of-squares 44.2%

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Applied egg-rr 44.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 38.7%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

   8. Taylor expanded in b around 0 18.1%

      \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

   9. Taylor expanded in a around 0 23.4%

      \[\leadsto \color{blue}{\left(a \cdot b\right)} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{+156}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 21.0% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b \cdot a\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* 0.011111111111111112 (* angle_m PI)) (* b a))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((0.011111111111111112 * (angle_m * ((double) M_PI))) * (b * a));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((0.011111111111111112 * (angle_m * Math.PI)) * (b * a));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((0.011111111111111112 * (angle_m * math.pi)) * (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(Float64(0.011111111111111112 * Float64(angle_m * 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 * 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[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 49.1%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 49.1%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 49.1%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 49.1%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 53.1%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 53.1%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in angle around 0 50.2%

   \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

8. Taylor expanded in b around 0 35.4%

   \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

9. Taylor expanded in a around 0 20.0%

   \[\leadsto \color{blue}{\left(a \cdot b\right)} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

10. Final simplification 20.0%

    \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot a\right) \]
11. Add Preprocessing

Alternative 23: 19.8% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(angle\_m \cdot a\right) \cdot \left(\pi \cdot b\right)\right)\right) \end{array} \]

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 a) (* PI b)))))

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 * a) * (((double) M_PI) * b)));
}

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 * a) * (Math.PI * b)));
}

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 * a) * (math.pi * b)))

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 * a) * Float64(pi * b))))
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 * a) * (pi * b)));
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 * a), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 49.1%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 49.1%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 49.1%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 49.1%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 53.1%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 53.1%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in angle around 0 50.2%

   \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

8. Taylor expanded in b around 0 35.4%

   \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

9. Taylor expanded in a around 0 17.8%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* 17.4%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)} \]

    2. *-commutative 17.4%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot b\right)}\right) \]

11. Simplified 17.4%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot b\right)\right)} \]

12. Final simplification 17.4%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot a\right) \cdot \left(\pi \cdot b\right)\right) \]
13. Add Preprocessing

Alternative 24: 19.9% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right)\right)\right) \end{array} \]

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 (* a (* angle_m (* PI b))))))

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 * (a * (angle_m * (((double) M_PI) * b))));
}

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 * (a * (angle_m * (Math.PI * b))));
}

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 * (a * (angle_m * (math.pi * b))))

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(a * Float64(angle_m * Float64(pi * b)))))
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 * (a * (angle_m * (pi * b))));
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[(a * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   2. *-commutative 49.1%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. associate-*l* 49.1%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]

3. Simplified 49.1%

   \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow2 49.1%

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. unpow2 49.1%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. difference-of-squares 53.1%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 53.1%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

7. Taylor expanded in angle around 0 50.2%

   \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \]

8. Taylor expanded in b around 0 35.4%

   \[\leadsto \left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \]

9. Taylor expanded in a around 0 17.8%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 17.8%

       \[\leadsto 0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot b\right)}\right)\right) \]

11. Simplified 17.8%

    \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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)))))