ab-angle->ABCF B

Percentage Accurate: 54.0% → 67.3%

Time: 46.3s

Alternatives: 9

Speedup: 32.2×

• 33.1% 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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 2 \cdot 10^{+217}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \sin \left(2 \cdot t\_0\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{elif}\;b\_m \leq 1.9 \cdot 10^{+264}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b\_m + a\right)\right) \cdot \left(\left(b\_m - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= b_m 2e+217)
      (* (* (- b_m a) (sin (* 2.0 t_0))) (+ b_m a))
      (if (<= b_m 1.9e+264)
        (* 0.011111111111111112 (* (* angle_m (+ b_m a)) (* (- b_m a) PI)))
        (*
         (* (* 2.0 (* (- b_m a) (+ b_m a))) (sin (expm1 (log1p t_0))))
         (cos (* PI (/ angle_m 180.0)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if (b_m <= 2e+217) {
		tmp = ((b_m - a) * sin((2.0 * t_0))) * (b_m + a);
	} else if (b_m <= 1.9e+264) {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * ((double) M_PI)));
	} else {
		tmp = ((2.0 * ((b_m - a) * (b_m + a))) * sin(expm1(log1p(t_0)))) * cos((((double) M_PI) * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double tmp;
	if (b_m <= 2e+217) {
		tmp = ((b_m - a) * Math.sin((2.0 * t_0))) * (b_m + a);
	} else if (b_m <= 1.9e+264) {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * Math.PI));
	} else {
		tmp = ((2.0 * ((b_m - a) * (b_m + a))) * Math.sin(Math.expm1(Math.log1p(t_0)))) * Math.cos((Math.PI * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	tmp = 0
	if b_m <= 2e+217:
		tmp = ((b_m - a) * math.sin((2.0 * t_0))) * (b_m + a)
	elif b_m <= 1.9e+264:
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * math.pi))
	else:
		tmp = ((2.0 * ((b_m - a) * (b_m + a))) * math.sin(math.expm1(math.log1p(t_0)))) * math.cos((math.pi * (angle_m / 180.0)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (b_m <= 2e+217)
		tmp = Float64(Float64(Float64(b_m - a) * sin(Float64(2.0 * t_0))) * Float64(b_m + a));
	elseif (b_m <= 1.9e+264)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b_m + a)) * Float64(Float64(b_m - a) * pi)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b_m - a) * Float64(b_m + a))) * sin(expm1(log1p(t_0)))) * cos(Float64(pi * Float64(angle_m / 180.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 2e+217], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.9e+264], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if b < 1.99999999999999992e217

   1. Initial program 55.7%

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

      1. associate-*l* 55.7%

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

      2. *-commutative 55.7%

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

      3. associate-*l* 55.7%

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

   3. Simplified 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

      3. difference-of-squares 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. clear-num 58.8%

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

      2. un-div-inv 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. associate-/r/ 58.3%

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

   10. Simplified 58.3%

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

       1. associate-*r* 58.3%

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

       2. associate-*l/ 57.4%

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

       3. associate-*r/ 59.2%

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

       4. div-inv 58.4%

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

       5. metadata-eval 58.4%

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

       6. rem-exp-log 29.2%

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

       7. associate-*r* 29.2%

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

       8. pow1 29.2%

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

   12. Applied egg-rr 69.9%

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

   if 1.99999999999999992e217 < b < 1.9000000000000001e264

   1. Initial program 12.1%

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

      1. associate-*l* 12.1%

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

      2. *-commutative 12.1%

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

      3. associate-*l* 12.1%

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

   3. Simplified 12.1%

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

      1. unpow2 12.1%

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

      2. unpow2 12.1%

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

      3. difference-of-squares 34.9%

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

   6. Applied egg-rr 34.9%

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

   7. Taylor expanded in angle around 0 57.1%

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

      1. associate-*r* 57.1%

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

      2. sub-neg 57.1%

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

      3. distribute-lft-in 45.4%

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

      4. +-commutative 45.4%

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

      5. +-commutative 45.4%

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

   9. Applied egg-rr 45.4%

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

       1. *-commutative 45.4%

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

       2. associate-*r* 45.4%

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

       3. *-commutative 45.4%

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

       4. associate-*r* 45.4%

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

       5. neg-mul-1 45.4%

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

       6. associate-*r* 45.4%

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

       7. distribute-rgt-out 57.1%

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

       8. +-commutative 57.1%

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

       9. associate-*r* 57.1%

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

       10. neg-mul-1 57.1%

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

       11. distribute-rgt-in 57.1%

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

       12. sub-neg 57.1%

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

   11. Simplified 57.1%

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

       1. pow1 57.1%

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

       2. associate-*r* 77.6%

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

       3. +-commutative 77.6%

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

   13. Applied egg-rr 77.6%

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

   if 1.9000000000000001e264 < b

   1. Initial program 50.0%

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

      1. unpow2 50.0%

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

      2. unpow2 50.0%

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

      3. difference-of-squares 62.5%

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

   4. Applied egg-rr 62.5%

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

      1. div-inv 50.0%

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

      2. metadata-eval 50.0%

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

      3. expm1-log1p-u 75.0%

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

   6. Applied egg-rr 75.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+217}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+264}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 2.3 \cdot 10^{+221}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{b\_m + a} \cdot \sqrt[3]{\left(b\_m - a\right) \cdot \sin \left(e^{\log \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 2.3e+221)
    (*
     (* (- b_m a) (sin (* 2.0 (* PI (* angle_m 0.005555555555555556)))))
     (+ b_m a))
    (pow
     (*
      (cbrt (+ b_m a))
      (cbrt
       (*
        (- b_m a)
        (sin (exp (log (* PI (* angle_m 0.011111111111111112))))))))
     3.0))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 2.3e+221) {
		tmp = ((b_m - a) * sin((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556))))) * (b_m + a);
	} else {
		tmp = pow((cbrt((b_m + a)) * cbrt(((b_m - a) * sin(exp(log((((double) M_PI) * (angle_m * 0.011111111111111112)))))))), 3.0);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 2.3e+221) {
		tmp = ((b_m - a) * Math.sin((2.0 * (Math.PI * (angle_m * 0.005555555555555556))))) * (b_m + a);
	} else {
		tmp = Math.pow((Math.cbrt((b_m + a)) * Math.cbrt(((b_m - a) * Math.sin(Math.exp(Math.log((Math.PI * (angle_m * 0.011111111111111112)))))))), 3.0);
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 2.3e+221)
		tmp = Float64(Float64(Float64(b_m - a) * sin(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556))))) * Float64(b_m + a));
	else
		tmp = Float64(cbrt(Float64(b_m + a)) * cbrt(Float64(Float64(b_m - a) * sin(exp(log(Float64(pi * Float64(angle_m * 0.011111111111111112)))))))) ^ 3.0;
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 2.3e+221], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(b$95$m + a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[Exp[N[Log[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 2.29999999999999987e221

   1. Initial program 55.7%

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

      1. associate-*l* 55.7%

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

      2. *-commutative 55.7%

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

      3. associate-*l* 55.7%

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

   3. Simplified 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

      3. difference-of-squares 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. clear-num 58.8%

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

      2. un-div-inv 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. associate-/r/ 58.3%

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

   10. Simplified 58.3%

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

       1. associate-*r* 58.3%

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

       2. associate-*l/ 57.4%

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

       3. associate-*r/ 59.2%

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

       4. div-inv 58.4%

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

       5. metadata-eval 58.4%

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

       6. rem-exp-log 29.2%

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

       7. associate-*r* 29.2%

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

       8. pow1 29.2%

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

   12. Applied egg-rr 69.9%

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

   if 2.29999999999999987e221 < b

   1. Initial program 29.9%

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

      1. associate-*l* 29.9%

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

      2. *-commutative 29.9%

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

      3. associate-*l* 29.9%

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

   3. Simplified 29.9%

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

      1. add-cube-cbrt 29.9%

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

      2. pow3 29.9%

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

      3. 2-sin 29.9%

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

      4. associate-*r* 29.9%

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

      5. div-inv 24.0%

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

      6. metadata-eval 24.0%

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

   6. Applied egg-rr 24.0%

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

      1. cbrt-prod 24.0%

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

      2. associate-*l* 24.0%

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

      3. metadata-eval 24.0%

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

      4. div-inv 29.9%

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

      5. 2-sin 29.9%

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

      6. unpow2 29.9%

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

      7. unpow2 29.9%

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

      8. difference-of-squares 47.9%

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

      9. cbrt-prod 47.9%

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

      10. associate-*l* 58.6%

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

   8. Applied egg-rr 64.5%

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

      1. +-commutative 64.5%

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

      2. associate-*l* 52.6%

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

   10. Simplified 52.6%

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

       1. add-exp-log 46.3%

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

   12. Applied egg-rr 46.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{+221}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{b + a} \cdot \sqrt[3]{\left(b - a\right) \cdot \sin \left(e^{\log \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 10^{-292}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right) \cdot \sin \left(2 \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b\_m + a\right)\right) \cdot \left(\left(b\_m - a\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow a 2.0) 1e-292)
    (*
     (* (- b_m a) (+ b_m a))
     (sin (* 2.0 (* angle_m (* PI 0.005555555555555556)))))
    (* 0.011111111111111112 (* (* angle_m (+ b_m a)) (* (- b_m a) PI))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (pow(a, 2.0) <= 1e-292) {
		tmp = ((b_m - a) * (b_m + a)) * sin((2.0 * (angle_m * (((double) M_PI) * 0.005555555555555556))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (Math.pow(a, 2.0) <= 1e-292) {
		tmp = ((b_m - a) * (b_m + a)) * Math.sin((2.0 * (angle_m * (Math.PI * 0.005555555555555556))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * Math.PI));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if math.pow(a, 2.0) <= 1e-292:
		tmp = ((b_m - a) * (b_m + a)) * math.sin((2.0 * (angle_m * (math.pi * 0.005555555555555556))))
	else:
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * math.pi))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if ((a ^ 2.0) <= 1e-292)
		tmp = Float64(Float64(Float64(b_m - a) * Float64(b_m + a)) * sin(Float64(2.0 * Float64(angle_m * Float64(pi * 0.005555555555555556)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b_m + a)) * Float64(Float64(b_m - a) * pi)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((a ^ 2.0) <= 1e-292)
		tmp = ((b_m - a) * (b_m + a)) * sin((2.0 * (angle_m * (pi * 0.005555555555555556))));
	else
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * pi));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 1e-292], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1.0000000000000001e-292

   1. Initial program 71.1%

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

      1. associate-*l* 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*l* 71.1%

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

   3. Simplified 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. difference-of-squares 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. clear-num 72.5%

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

      2. un-div-inv 69.4%

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

   8. Applied egg-rr 69.4%

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

      1. associate-/r/ 69.9%

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

   10. Simplified 69.9%

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

       1. associate-*r* 69.9%

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

       2. associate-*l/ 68.7%

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

       3. associate-*r/ 71.1%

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

       4. div-inv 71.0%

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

       5. metadata-eval 71.0%

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

       6. rem-exp-log 37.0%

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

       7. associate-*r* 37.0%

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

       8. pow1 37.0%

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

   12. Applied egg-rr 72.4%

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

       1. unpow1 72.4%

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

       2. associate-*r* 71.1%

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

       3. *-commutative 71.1%

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

       4. +-commutative 71.1%

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

       5. *-commutative 71.1%

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

       6. associate-*l* 70.0%

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

       7. *-commutative 70.0%

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

   14. Simplified 70.0%

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

   if 1.0000000000000001e-292 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 46.3%

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

      1. associate-*l* 46.3%

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

      2. *-commutative 46.3%

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

      3. associate-*l* 46.3%

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

   3. Simplified 46.3%

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

      1. unpow2 46.3%

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

      2. unpow2 46.3%

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

      3. difference-of-squares 52.7%

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

   6. Applied egg-rr 52.7%

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

   7. Taylor expanded in angle around 0 50.9%

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

      1. associate-*r* 50.9%

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

      2. sub-neg 50.9%

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

      3. distribute-lft-in 45.5%

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

      4. +-commutative 45.5%

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

      5. +-commutative 45.5%

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

   9. Applied egg-rr 45.5%

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

       1. *-commutative 45.5%

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

       2. associate-*r* 45.5%

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

       3. *-commutative 45.5%

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

       4. associate-*r* 45.5%

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

       5. neg-mul-1 45.5%

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

       6. associate-*r* 45.5%

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

       7. distribute-rgt-out 50.9%

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

       8. +-commutative 50.9%

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

       9. associate-*r* 50.9%

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

       10. neg-mul-1 50.9%

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

       11. distribute-rgt-in 50.9%

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

       12. sub-neg 50.9%

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

   11. Simplified 50.9%

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

       1. pow1 50.9%

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

       2. associate-*r* 65.9%

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

       3. +-commutative 65.9%

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

   13. Applied egg-rr 65.9%

       \[\leadsto \color{blue}{{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)}^{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 10^{-292}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\right) \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.8 \cdot 10^{+64}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b\_m + a\right)\right) \cdot t\_0\right)\\ \mathbf{elif}\;angle\_m \leq 2.2 \cdot 10^{+270}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m + a\right) \cdot \left(a \cdot \left(-\pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m + a\right) \cdot \left|t\_0\right|\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a) PI)))
   (*
    angle_s
    (if (<= angle_m 3.8e+64)
      (* 0.011111111111111112 (* (* angle_m (+ b_m a)) t_0))
      (if (<= angle_m 2.2e+270)
        (* 0.011111111111111112 (* angle_m (* (+ b_m a) (* a (- PI)))))
        (* 0.011111111111111112 (* angle_m (* (+ b_m a) (fabs t_0)))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * ((double) M_PI);
	double tmp;
	if (angle_m <= 3.8e+64) {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * t_0);
	} else if (angle_m <= 2.2e+270) {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * fabs(t_0)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (b_m - a) * Math.PI;
	double tmp;
	if (angle_m <= 3.8e+64) {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * t_0);
	} else if (angle_m <= 2.2e+270) {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * Math.abs(t_0)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (b_m - a) * math.pi
	tmp = 0
	if angle_m <= 3.8e+64:
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * t_0)
	elif angle_m <= 2.2e+270:
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * math.fabs(t_0)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a) * pi)
	tmp = 0.0
	if (angle_m <= 3.8e+64)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b_m + a)) * t_0));
	elseif (angle_m <= 2.2e+270)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b_m + a) * Float64(a * Float64(-pi)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b_m + a) * abs(t_0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (b_m - a) * pi;
	tmp = 0.0;
	if (angle_m <= 3.8e+64)
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * t_0);
	elseif (angle_m <= 2.2e+270)
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * abs(t_0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a), $MachinePrecision] * Pi), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 3.8e+64], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.2e+270], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b$95$m + a), $MachinePrecision] * N[(a * (-Pi)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b$95$m + a), $MachinePrecision] * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.8000000000000001e64

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. *-commutative 58.5%

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

      3. associate-*l* 58.5%

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

   3. Simplified 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. difference-of-squares 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in angle around 0 61.4%

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

      1. associate-*r* 61.4%

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

      2. sub-neg 61.4%

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

      3. distribute-lft-in 57.3%

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

      4. +-commutative 57.3%

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

      5. +-commutative 57.3%

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

   9. Applied egg-rr 57.3%

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

       1. *-commutative 57.3%

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

       2. associate-*r* 57.3%

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

       3. *-commutative 57.3%

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

       4. associate-*r* 57.3%

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

       5. neg-mul-1 57.3%

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

       6. associate-*r* 57.3%

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

       7. distribute-rgt-out 61.4%

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

       8. +-commutative 61.4%

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

       9. associate-*r* 61.4%

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

       10. neg-mul-1 61.4%

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

       11. distribute-rgt-in 61.4%

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

       12. sub-neg 61.4%

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

   11. Simplified 61.4%

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

       1. pow1 61.4%

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

       2. associate-*r* 73.7%

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

       3. +-commutative 73.7%

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

   13. Applied egg-rr 73.7%

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

   if 3.8000000000000001e64 < angle < 2.2000000000000001e270

   1. Initial program 30.4%

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

      1. associate-*l* 30.4%

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

      2. *-commutative 30.4%

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

      3. associate-*l* 30.4%

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

   3. Simplified 30.4%

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

      1. unpow2 30.4%

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

      2. unpow2 30.4%

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

      3. difference-of-squares 33.0%

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

   6. Applied egg-rr 33.0%

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

   7. Taylor expanded in angle around 0 20.4%

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

      1. associate-*r* 20.4%

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

      2. sub-neg 20.4%

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

      3. distribute-lft-in 17.8%

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

      4. +-commutative 17.8%

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

      5. +-commutative 17.8%

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

   9. Applied egg-rr 17.8%

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

       1. *-commutative 17.8%

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

       2. associate-*r* 17.8%

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

       3. *-commutative 17.8%

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

       4. associate-*r* 17.8%

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

       5. neg-mul-1 17.8%

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

       6. associate-*r* 17.8%

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

       7. distribute-rgt-out 20.4%

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

       8. +-commutative 20.4%

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

       9. associate-*r* 20.4%

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

       10. neg-mul-1 20.4%

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

       11. distribute-rgt-in 20.4%

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

       12. sub-neg 20.4%

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

   11. Simplified 20.4%

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

   12. Taylor expanded in b around 0 21.3%

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

       1. associate-*r* 21.3%

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

       2. mul-1-neg 21.3%

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

   14. Simplified 21.3%

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

   if 2.2000000000000001e270 < angle

   1. Initial program 51.6%

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

      1. associate-*l* 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l* 51.6%

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

   3. Simplified 51.6%

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

      1. unpow2 51.6%

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

      2. unpow2 51.6%

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

      3. difference-of-squares 51.6%

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in angle around 0 50.3%

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

      1. associate-*r* 50.3%

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

      2. sub-neg 50.3%

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

      3. distribute-lft-in 50.3%

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

      4. +-commutative 50.3%

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

      5. +-commutative 50.3%

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

   9. Applied egg-rr 50.3%

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

       1. *-commutative 50.3%

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

       2. associate-*r* 50.3%

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

       3. *-commutative 50.3%

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

       4. associate-*r* 50.3%

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

       5. neg-mul-1 50.3%

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

       6. associate-*r* 50.3%

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

       7. distribute-rgt-out 50.3%

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

       8. +-commutative 50.3%

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

       9. associate-*r* 50.3%

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

       10. neg-mul-1 50.3%

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

       11. distribute-rgt-in 50.3%

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

       12. sub-neg 50.3%

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

   11. Simplified 50.3%

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

       1. add-sqr-sqrt 17.0%

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

       2. sqrt-unprod 25.6%

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

       3. pow2 25.6%

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

   13. Applied egg-rr 25.6%

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

       1. unpow2 25.6%

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

       2. rem-sqrt-square 25.6%

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

   15. Simplified 25.6%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.8 \cdot 10^{+64}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{elif}\;angle \leq 2.2 \cdot 10^{+270}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(a \cdot \left(-\pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left|\left(b - a\right) \cdot \pi\right|\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{-145}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b\_m + a\right)\right) \cdot \left(\left(b\_m - a\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 3e-145)
    (*
     (sin (* 2.0 (* PI (* angle_m 0.005555555555555556))))
     (* (- b_m a) (+ b_m a)))
    (* 0.011111111111111112 (* (* angle_m (+ b_m a)) (* (- b_m a) PI))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 3e-145) {
		tmp = sin((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556)))) * ((b_m - a) * (b_m + a));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 3e-145) {
		tmp = Math.sin((2.0 * (Math.PI * (angle_m * 0.005555555555555556)))) * ((b_m - a) * (b_m + a));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * Math.PI));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 3e-145:
		tmp = math.sin((2.0 * (math.pi * (angle_m * 0.005555555555555556)))) * ((b_m - a) * (b_m + a))
	else:
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * math.pi))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 3e-145)
		tmp = Float64(sin(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556)))) * Float64(Float64(b_m - a) * Float64(b_m + a)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b_m + a)) * Float64(Float64(b_m - a) * pi)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 3e-145)
		tmp = sin((2.0 * (pi * (angle_m * 0.005555555555555556)))) * ((b_m - a) * (b_m + a));
	else
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * pi));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 3e-145], N[(N[Sin[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.99999999999999992e-145

   1. Initial program 56.3%

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

      1. associate-*l* 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*l* 56.3%

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

   3. Simplified 56.3%

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

      1. unpow2 56.3%

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

      2. unpow2 56.3%

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

      3. difference-of-squares 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. clear-num 59.9%

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

      2. un-div-inv 59.5%

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

   8. Applied egg-rr 59.5%

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

      1. associate-/r/ 60.0%

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

   10. Simplified 60.0%

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

       1. associate-*r* 60.0%

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

       2. associate-*l/ 58.7%

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

       3. associate-*r/ 59.9%

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

       4. div-inv 59.6%

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

       5. metadata-eval 59.6%

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

       6. rem-exp-log 33.0%

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

       7. associate-*r* 33.0%

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

       8. pow1 33.0%

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

   12. Applied egg-rr 70.3%

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

       1. unpow1 70.3%

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

       2. associate-*r* 60.6%

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

       3. *-commutative 60.6%

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

       4. +-commutative 60.6%

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

   14. Simplified 60.6%

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

   if 2.99999999999999992e-145 < a

   1. Initial program 49.4%

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

      1. associate-*l* 49.4%

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

      2. *-commutative 49.4%

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

      3. associate-*l* 49.4%

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

   3. Simplified 49.4%

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

      1. unpow2 49.4%

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

      2. unpow2 49.4%

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

      3. difference-of-squares 55.4%

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

   6. Applied egg-rr 55.4%

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

   7. Taylor expanded in angle around 0 52.0%

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

      1. associate-*r* 52.0%

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

      2. sub-neg 52.0%

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

      3. distribute-lft-in 46.0%

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

      4. +-commutative 46.0%

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

      5. +-commutative 46.0%

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

   9. Applied egg-rr 46.0%

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

       1. *-commutative 46.0%

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

       2. associate-*r* 46.0%

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

       3. *-commutative 46.0%

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

       4. associate-*r* 46.0%

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

       5. neg-mul-1 46.0%

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

       6. associate-*r* 46.0%

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

       7. distribute-rgt-out 52.0%

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

       8. +-commutative 52.0%

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

       9. associate-*r* 52.0%

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

       10. neg-mul-1 52.0%

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

       11. distribute-rgt-in 52.0%

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

       12. sub-neg 52.0%

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

   11. Simplified 52.0%

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

       1. pow1 52.0%

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

       2. associate-*r* 64.9%

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

       3. +-commutative 64.9%

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

   13. Applied egg-rr 64.9%

       \[\leadsto \color{blue}{{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)}^{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{-145}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 2 \cdot 10^{+217}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b\_m + a\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b\_m + a\right)\right) \cdot \left(\left(b\_m - a\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 2e+217)
    (*
     (* (- b_m a) (sin (* 2.0 (* PI (* angle_m 0.005555555555555556)))))
     (+ b_m a))
    (* 0.011111111111111112 (* (* angle_m (+ b_m a)) (* (- b_m a) PI))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 2e+217) {
		tmp = ((b_m - a) * sin((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556))))) * (b_m + a);
	} else {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 2e+217) {
		tmp = ((b_m - a) * Math.sin((2.0 * (Math.PI * (angle_m * 0.005555555555555556))))) * (b_m + a);
	} else {
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * Math.PI));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if b_m <= 2e+217:
		tmp = ((b_m - a) * math.sin((2.0 * (math.pi * (angle_m * 0.005555555555555556))))) * (b_m + a)
	else:
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * math.pi))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 2e+217)
		tmp = Float64(Float64(Float64(b_m - a) * sin(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556))))) * Float64(b_m + a));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b_m + a)) * Float64(Float64(b_m - a) * pi)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (b_m <= 2e+217)
		tmp = ((b_m - a) * sin((2.0 * (pi * (angle_m * 0.005555555555555556))))) * (b_m + a);
	else
		tmp = 0.011111111111111112 * ((angle_m * (b_m + a)) * ((b_m - a) * pi));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 2e+217], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.99999999999999992e217

   1. Initial program 55.7%

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

      1. associate-*l* 55.7%

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

      2. *-commutative 55.7%

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

      3. associate-*l* 55.7%

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

   3. Simplified 55.7%

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

      1. unpow2 55.7%

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

      2. unpow2 55.7%

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

      3. difference-of-squares 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. clear-num 58.8%

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

      2. un-div-inv 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. associate-/r/ 58.3%

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

   10. Simplified 58.3%

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

       1. associate-*r* 58.3%

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

       2. associate-*l/ 57.4%

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

       3. associate-*r/ 59.2%

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

       4. div-inv 58.4%

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

       5. metadata-eval 58.4%

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

       6. rem-exp-log 29.2%

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

       7. associate-*r* 29.2%

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

       8. pow1 29.2%

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

   12. Applied egg-rr 69.9%

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

   if 1.99999999999999992e217 < b

   1. Initial program 29.9%

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

      1. associate-*l* 29.9%

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

      2. *-commutative 29.9%

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

      3. associate-*l* 29.9%

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

   3. Simplified 29.9%

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

      1. unpow2 29.9%

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

      2. unpow2 29.9%

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

      3. difference-of-squares 47.9%

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

   6. Applied egg-rr 47.9%

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

   7. Taylor expanded in angle around 0 59.6%

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

      1. associate-*r* 59.6%

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

      2. sub-neg 59.6%

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

      3. distribute-lft-in 47.6%

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

      4. +-commutative 47.6%

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

      5. +-commutative 47.6%

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

   9. Applied egg-rr 47.6%

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

       1. *-commutative 47.6%

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

       2. associate-*r* 47.6%

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

       3. *-commutative 47.6%

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

       4. associate-*r* 47.6%

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

       5. neg-mul-1 47.6%

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

       6. associate-*r* 47.6%

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

       7. distribute-rgt-out 59.6%

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

       8. +-commutative 59.6%

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

       9. associate-*r* 59.6%

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

       10. neg-mul-1 59.6%

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

       11. distribute-rgt-in 59.6%

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

       12. sub-neg 59.6%

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

   11. Simplified 59.6%

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

       1. pow1 59.6%

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

       2. associate-*r* 70.5%

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

       3. +-commutative 70.5%

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

   13. Applied egg-rr 70.5%

       \[\leadsto \color{blue}{{\left(0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)}^{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+217}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.5% accurate, 24.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 7.6 \cdot 10^{-7}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b\_m + a\right) \cdot \left(a \cdot \left(-\pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 7.6e-7)
    (* 0.011111111111111112 (* angle_m (* (+ b_m a) (* b_m PI))))
    (* 0.011111111111111112 (* angle_m (* (+ b_m a) (* a (- PI))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 7.6e-7) {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (b_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 7.6e-7) {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (b_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -Math.PI)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 7.6e-7:
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (b_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -math.pi)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 7.6e-7)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b_m + a) * Float64(b_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b_m + a) * Float64(a * Float64(-pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 7.6e-7)
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (b_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b_m + a) * (a * -pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 7.6e-7], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b$95$m + a), $MachinePrecision] * N[(a * (-Pi)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 7.60000000000000029e-7

   1. Initial program 55.4%

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

      1. associate-*l* 55.4%

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

      2. *-commutative 55.4%

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

      3. associate-*l* 55.4%

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

   3. Simplified 55.4%

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

      1. unpow2 55.4%

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

      2. unpow2 55.4%

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

      3. difference-of-squares 58.6%

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

   6. Applied egg-rr 58.6%

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

   7. Taylor expanded in angle around 0 55.6%

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

      1. associate-*r* 55.6%

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

      2. sub-neg 55.6%

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

      3. distribute-lft-in 53.3%

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

      4. +-commutative 53.3%

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

      5. +-commutative 53.3%

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

   9. Applied egg-rr 53.3%

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

       1. *-commutative 53.3%

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

       2. associate-*r* 53.3%

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

       3. *-commutative 53.3%

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

       4. associate-*r* 53.3%

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

       5. neg-mul-1 53.3%

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

       6. associate-*r* 53.3%

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

       7. distribute-rgt-out 55.6%

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

       8. +-commutative 55.6%

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

       9. associate-*r* 55.6%

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

       10. neg-mul-1 55.6%

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

       11. distribute-rgt-in 55.6%

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

       12. sub-neg 55.6%

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

   11. Simplified 55.6%

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

   12. Taylor expanded in b around inf 42.8%

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

       1. *-commutative 42.8%

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

   14. Simplified 42.8%

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

   if 7.60000000000000029e-7 < a

   1. Initial program 50.0%

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

      1. associate-*l* 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*l* 50.0%

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

   3. Simplified 50.0%

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

      1. unpow2 50.0%

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

      2. unpow2 50.0%

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

      3. difference-of-squares 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in angle around 0 52.3%

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

      1. associate-*r* 52.3%

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

      2. sub-neg 52.3%

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

      3. distribute-lft-in 44.4%

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

      4. +-commutative 44.4%

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

      5. +-commutative 44.4%

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

   9. Applied egg-rr 44.4%

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

       1. *-commutative 44.4%

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

       2. associate-*r* 44.4%

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

       3. *-commutative 44.4%

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

       4. associate-*r* 44.4%

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

       5. neg-mul-1 44.4%

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

       6. associate-*r* 44.4%

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

       7. distribute-rgt-out 52.3%

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

       8. +-commutative 52.3%

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

       9. associate-*r* 52.3%

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

       10. neg-mul-1 52.3%

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

       11. distribute-rgt-in 52.3%

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

       12. sub-neg 52.3%

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

   11. Simplified 52.3%

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

   12. Taylor expanded in b around 0 48.0%

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

       1. associate-*r* 48.0%

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

       2. mul-1-neg 48.0%

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

   14. Simplified 48.0%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.6 \cdot 10^{-7}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \left(a \cdot \left(-\pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.1% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. associate-*l* 54.0%

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

   2. *-commutative 54.0%

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

   3. associate-*l* 54.0%

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

3. Simplified 54.0%

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

   1. unpow2 54.0%

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

   2. unpow2 54.0%

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

   3. difference-of-squares 58.4%

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

6. Applied egg-rr 58.4%

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

7. Taylor expanded in angle around 0 54.8%

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

8. Final simplification 54.8%

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

Alternative 9: 37.1% accurate, 38.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. associate-*l* 54.0%

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

   2. *-commutative 54.0%

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

   3. associate-*l* 54.0%

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

3. Simplified 54.0%

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

   1. unpow2 54.0%

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

   2. unpow2 54.0%

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

   3. difference-of-squares 58.4%

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

6. Applied egg-rr 58.4%

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

7. Taylor expanded in angle around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. sub-neg 54.8%

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

   3. distribute-lft-in 51.1%

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

   4. +-commutative 51.1%

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

   5. +-commutative 51.1%

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

9. Applied egg-rr 51.1%

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

    1. *-commutative 51.1%

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

    2. associate-*r* 51.1%

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

    3. *-commutative 51.1%

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

    4. associate-*r* 51.1%

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

    5. neg-mul-1 51.1%

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

    6. associate-*r* 51.1%

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

    7. distribute-rgt-out 54.8%

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

    8. +-commutative 54.8%

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

    9. associate-*r* 54.8%

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

    10. neg-mul-1 54.8%

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

    11. distribute-rgt-in 54.8%

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

    12. sub-neg 54.8%

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

11. Simplified 54.8%

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

12. Taylor expanded in b around inf 37.7%

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

    1. *-commutative 37.7%

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

14. Simplified 37.7%

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

15. Final simplification 37.7%

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

Reproduce

herbie shell --seed 2024067 
(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)))))