ab-angle->ABCF B

Percentage Accurate: 53.7% → 65.9%

Time: 17.0s

Alternatives: 22

Speedup: 8.5×

• 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: 53.7% 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: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ t_1 := \mathsf{fma}\left(angle\_m, \frac{180}{angle\_m}, -180\right)\\ t_2 := \frac{32400 \cdot \left(\pi \cdot \pi\right)}{t\_1 \cdot \left(-\pi\right)}\\ t_3 := \frac{180}{angle\_m \cdot 0.005555555555555556}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+133}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(angle\_m \cdot \pi\right)\right)}{angle\_m \cdot angle\_m}, \frac{1}{\pi \cdot t\_1}, t\_2\right)}{t\_3}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{\frac{180}{angle\_m}}{\pi}, \frac{\left(angle\_m \cdot \pi\right) \cdot \frac{angle\_m \cdot \pi}{angle\_m \cdot 0.005555555555555556}}{t\_1}, t\_2\right)}{t\_3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* angle_m PI) 0.011111111111111112)))
        (t_1 (fma angle_m (/ 180.0 angle_m) -180.0))
        (t_2 (/ (* 32400.0 (* PI PI)) (* t_1 (- PI))))
        (t_3 (/ 180.0 (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+25)
      (* (+ b a) (* (- b a) t_0))
      (if (<= (/ angle_m 180.0) 1e+133)
        (*
         (+ b a)
         (*
          (- b a)
          (sin
           (/
            (fma
             (/
              (* 32400.0 (* (* angle_m PI) (* angle_m PI)))
              (* angle_m angle_m))
             (/ 1.0 (* PI t_1))
             t_2)
            t_3))))
        (if (<= (/ angle_m 180.0) 5e+226)
          (*
           (+ b a)
           (*
            (- b a)
            (sin
             (/
              (fma
               (/ (/ 180.0 angle_m) PI)
               (/
                (*
                 (* angle_m PI)
                 (/ (* angle_m PI) (* angle_m 0.005555555555555556)))
                t_1)
               t_2)
              t_3))))
          (* (* a a) (* t_0 (fma b (/ b (* a a)) -1.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	double t_1 = fma(angle_m, (180.0 / angle_m), -180.0);
	double t_2 = (32400.0 * (((double) M_PI) * ((double) M_PI))) / (t_1 * -((double) M_PI));
	double t_3 = 180.0 / (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+25) {
		tmp = (b + a) * ((b - a) * t_0);
	} else if ((angle_m / 180.0) <= 1e+133) {
		tmp = (b + a) * ((b - a) * sin((fma(((32400.0 * ((angle_m * ((double) M_PI)) * (angle_m * ((double) M_PI)))) / (angle_m * angle_m)), (1.0 / (((double) M_PI) * t_1)), t_2) / t_3)));
	} else if ((angle_m / 180.0) <= 5e+226) {
		tmp = (b + a) * ((b - a) * sin((fma(((180.0 / angle_m) / ((double) M_PI)), (((angle_m * ((double) M_PI)) * ((angle_m * ((double) M_PI)) / (angle_m * 0.005555555555555556))) / t_1), t_2) / t_3)));
	} else {
		tmp = (a * a) * (t_0 * fma(b, (b / (a * a)), -1.0));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))
	t_1 = fma(angle_m, Float64(180.0 / angle_m), -180.0)
	t_2 = Float64(Float64(32400.0 * Float64(pi * pi)) / Float64(t_1 * Float64(-pi)))
	t_3 = Float64(180.0 / Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+25)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * t_0));
	elseif (Float64(angle_m / 180.0) <= 1e+133)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(fma(Float64(Float64(32400.0 * Float64(Float64(angle_m * pi) * Float64(angle_m * pi))) / Float64(angle_m * angle_m)), Float64(1.0 / Float64(pi * t_1)), t_2) / t_3))));
	elseif (Float64(angle_m / 180.0) <= 5e+226)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(fma(Float64(Float64(180.0 / angle_m) / pi), Float64(Float64(Float64(angle_m * pi) * Float64(Float64(angle_m * pi) / Float64(angle_m * 0.005555555555555556))) / t_1), t_2) / t_3))));
	else
		tmp = Float64(Float64(a * a) * Float64(t_0 * fma(b, Float64(b / Float64(a * a)), -1.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(180.0 / angle$95$m), $MachinePrecision] + -180.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(32400.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * (-Pi)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(180.0 / N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+25], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+133], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(N[(32400.0 * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+226], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(N[(180.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision] * N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] / N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(t$95$0 * N[(b * N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000018e25

   1. Initial program 56.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 75.8%

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

   if 2.00000000000000018e25 < (/.f64 angle #s(literal 180 binary64)) < 1e133

   1. Initial program 30.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 31.1%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. frac-add N/A

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

      13. /-lowering-/.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}}{180 \cdot \frac{180}{angle}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \color{blue}{\frac{180}{angle}}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, \color{blue}{180 \cdot \mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      20. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{180 \cdot \frac{180}{angle}}}\right)\right) \]

      22. /-lowering-/.f64 31.5

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

   6. Applied egg-rr 31.5%

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

      1. clear-num N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \color{blue}{\frac{1}{\frac{angle}{180}}}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      4. div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      6. metadata-eval 36.8

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

   8. Applied egg-rr 36.8%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle}\right), \frac{1}{\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot \mathsf{PI}\left(\right)}, \mathsf{neg}\left(\frac{\left(180 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(180 \cdot \mathsf{PI}\left(\right)\right)}{\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot \mathsf{PI}\left(\right)}\right)\right)}}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

   10. Applied egg-rr 54.5%

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)}{angle \cdot angle}, \frac{1}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, -\frac{\left(\pi \cdot \pi\right) \cdot 32400}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}\right)}}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right) \]

   if 1e133 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000005e226

   1. Initial program 29.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 33.1%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. frac-add N/A

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

      13. /-lowering-/.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}}{180 \cdot \frac{180}{angle}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \color{blue}{\frac{180}{angle}}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, \color{blue}{180 \cdot \mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      20. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{180 \cdot \frac{180}{angle}}}\right)\right) \]

      22. /-lowering-/.f64 39.2

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

   6. Applied egg-rr 39.2%

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

      1. clear-num N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \color{blue}{\frac{1}{\frac{angle}{180}}}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      4. div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      6. metadata-eval 40.4

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

   8. Applied egg-rr 40.4%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

   10. Applied egg-rr 51.3%

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{180}{angle}}{\pi}, \frac{\left(\pi \cdot angle\right) \cdot \frac{\pi \cdot angle}{angle \cdot 0.005555555555555556}}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, -\frac{\left(\pi \cdot \pi\right) \cdot 32400}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}\right)}}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right) \]

   if 5.0000000000000005e226 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 10.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\frac{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b + -1 \cdot b\right)}{a} + \frac{{b}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{a}^{2}}\right)\right)} \]

   7. Simplified 70.3%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+133}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)}{angle \cdot angle}, \frac{1}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, \frac{32400 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right) \cdot \left(-\pi\right)}\right)}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{\frac{180}{angle}}{\pi}, \frac{\left(angle \cdot \pi\right) \cdot \frac{angle \cdot \pi}{angle \cdot 0.005555555555555556}}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, \frac{32400 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right) \cdot \left(-\pi\right)}\right)}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(\pi \cdot \pi\right)\\ t_1 := {b}^{2} - {a}^{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;a \cdot \left(angle\_m \cdot \left(a \cdot \mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot t\_0\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-269}:\\ \;\;\;\;\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot t\_0, 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* PI PI))) (t_1 (- (pow b 2.0) (pow a 2.0))))
   (*
    angle_s
    (if (<= t_1 (- INFINITY))
      (*
       a
       (*
        angle_m
        (*
         a
         (fma
          (* angle_m (* angle_m t_0))
          2.2862368541380886e-7
          (* PI -0.011111111111111112)))))
      (if (<= t_1 5e-269)
        (* (sin (* (* angle_m PI) 0.011111111111111112)) (* a (- a)))
        (*
         (+ b a)
         (*
          angle_m
          (fma
           (* (* angle_m angle_m) -2.2862368541380886e-7)
           (* (- b a) t_0)
           (* 0.011111111111111112 (* (- b a) PI))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (((double) M_PI) * ((double) M_PI));
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = a * (angle_m * (a * fma((angle_m * (angle_m * t_0)), 2.2862368541380886e-7, (((double) M_PI) * -0.011111111111111112))));
	} else if (t_1 <= 5e-269) {
		tmp = sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)) * (a * -a);
	} else {
		tmp = (b + a) * (angle_m * fma(((angle_m * angle_m) * -2.2862368541380886e-7), ((b - a) * t_0), (0.011111111111111112 * ((b - a) * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(pi * pi))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(a * Float64(angle_m * Float64(a * fma(Float64(angle_m * Float64(angle_m * t_0)), 2.2862368541380886e-7, Float64(pi * -0.011111111111111112)))));
	elseif (t_1 <= 5e-269)
		tmp = Float64(sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)) * Float64(a * Float64(-a)));
	else
		tmp = Float64(Float64(b + a) * Float64(angle_m * fma(Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7), Float64(Float64(b - a) * t_0), Float64(0.011111111111111112 * Float64(Float64(b - a) * pi)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(a * N[(angle$95$m * N[(a * N[(N[(angle$95$m * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * 2.2862368541380886e-7 + N[(Pi * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-269], N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * N[(a * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 44.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 68.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{2}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(-1 \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(-1 \cdot {a}^{2}\right)} \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot a}\right)\right) \]

      13. *-lowering-*.f64 48.6

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

   7. Simplified 48.6%

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

      1. distribute-lft-neg-in N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot a\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      4. add-cbrt-cube N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      5. add-sqr-sqrt N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      6. associate-*r* N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      7. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      8. unpow1/3 N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \left(\color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      9. *-commutative N/A

         \[\leadsto \left(\sin \left(\color{blue}{\left(\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot angle\right)} \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

   9. Applied egg-rr 66.5%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \cdot a \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \left(a \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right)\right)\right) \cdot a \]

       2. associate-*r* N/A

          \[\leadsto \left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right) \cdot a \]

       3. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\left(\frac{1}{4374000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}}\right)\right) \cdot a \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{4374000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)} \cdot a \]

       5. +-commutative N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(\left(\frac{1}{4374000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot a \]

       6. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\frac{1}{4374000} \cdot \left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

       7. associate-*r* N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{4374000} \cdot \color{blue}{\left(a \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

       8. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{4374000} \cdot \left(a \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

       9. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \frac{1}{4374000}} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

   12. Simplified 72.5%

       \[\leadsto \color{blue}{\left(angle \cdot \left(a \cdot \mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\right)} \cdot a \]

   if -inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.99999999999999979e-269

   1. Initial program 63.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 65.7%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{2}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(-1 \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(-1 \cdot {a}^{2}\right)} \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot a}\right)\right) \]

      13. *-lowering-*.f64 64.9

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

   7. Simplified 64.9%

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

   if 4.99999999999999979e-269 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 46.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 69.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Simplified 70.1%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -\infty:\\ \;\;\;\;a \cdot \left(angle \cdot \left(a \cdot \mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{-269}:\\ \;\;\;\;\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -inf.0

   1. Initial program 54.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 43.7

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

   5. Simplified 43.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 25.7

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

   8. Simplified 25.7%

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

      1. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right) \]

      7. PI-lowering-PI.f64 32.0

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

   10. Applied egg-rr 32.0%

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot \left(\frac{-1}{90} \cdot a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       7. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(a \cdot \frac{-1}{90}\right)} \]

       8. *-lowering-*.f64 32.0

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

   12. Applied egg-rr 32.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   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. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 52.9

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

   5. Simplified 52.9%

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

4. Final simplification 49.2%

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

Alternative 4: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ t_1 := \mathsf{fma}\left(angle\_m, \frac{180}{angle\_m}, -180\right)\\ t_2 := \frac{32400 \cdot \left(\pi \cdot \pi\right)}{t\_1 \cdot \left(-\pi\right)}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t\_0\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+133}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(angle\_m \cdot \pi\right)\right)}{angle\_m \cdot angle\_m}, \frac{1}{\pi \cdot t\_1}, t\_2\right)}{\frac{180}{angle\_m \cdot 0.005555555555555556}}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{\frac{180}{angle\_m}}{\pi}, \frac{\left(angle\_m \cdot \pi\right) \cdot \frac{angle\_m \cdot \pi}{angle\_m \cdot 0.005555555555555556}}{t\_1}, t\_2\right)}{180 \cdot \frac{180}{angle\_m}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* angle_m PI) 0.011111111111111112)))
        (t_1 (fma angle_m (/ 180.0 angle_m) -180.0))
        (t_2 (/ (* 32400.0 (* PI PI)) (* t_1 (- PI)))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+25)
      (* (+ b a) (* (- b a) t_0))
      (if (<= (/ angle_m 180.0) 1e+133)
        (*
         (+ b a)
         (*
          (- b a)
          (sin
           (/
            (fma
             (/
              (* 32400.0 (* (* angle_m PI) (* angle_m PI)))
              (* angle_m angle_m))
             (/ 1.0 (* PI t_1))
             t_2)
            (/ 180.0 (* angle_m 0.005555555555555556))))))
        (if (<= (/ angle_m 180.0) 5e+226)
          (*
           (+ b a)
           (*
            (- b a)
            (sin
             (/
              (fma
               (/ (/ 180.0 angle_m) PI)
               (/
                (*
                 (* angle_m PI)
                 (/ (* angle_m PI) (* angle_m 0.005555555555555556)))
                t_1)
               t_2)
              (* 180.0 (/ 180.0 angle_m))))))
          (* (* a a) (* t_0 (fma b (/ b (* a a)) -1.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	double t_1 = fma(angle_m, (180.0 / angle_m), -180.0);
	double t_2 = (32400.0 * (((double) M_PI) * ((double) M_PI))) / (t_1 * -((double) M_PI));
	double tmp;
	if ((angle_m / 180.0) <= 2e+25) {
		tmp = (b + a) * ((b - a) * t_0);
	} else if ((angle_m / 180.0) <= 1e+133) {
		tmp = (b + a) * ((b - a) * sin((fma(((32400.0 * ((angle_m * ((double) M_PI)) * (angle_m * ((double) M_PI)))) / (angle_m * angle_m)), (1.0 / (((double) M_PI) * t_1)), t_2) / (180.0 / (angle_m * 0.005555555555555556)))));
	} else if ((angle_m / 180.0) <= 5e+226) {
		tmp = (b + a) * ((b - a) * sin((fma(((180.0 / angle_m) / ((double) M_PI)), (((angle_m * ((double) M_PI)) * ((angle_m * ((double) M_PI)) / (angle_m * 0.005555555555555556))) / t_1), t_2) / (180.0 * (180.0 / angle_m)))));
	} else {
		tmp = (a * a) * (t_0 * fma(b, (b / (a * a)), -1.0));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))
	t_1 = fma(angle_m, Float64(180.0 / angle_m), -180.0)
	t_2 = Float64(Float64(32400.0 * Float64(pi * pi)) / Float64(t_1 * Float64(-pi)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+25)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * t_0));
	elseif (Float64(angle_m / 180.0) <= 1e+133)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(fma(Float64(Float64(32400.0 * Float64(Float64(angle_m * pi) * Float64(angle_m * pi))) / Float64(angle_m * angle_m)), Float64(1.0 / Float64(pi * t_1)), t_2) / Float64(180.0 / Float64(angle_m * 0.005555555555555556))))));
	elseif (Float64(angle_m / 180.0) <= 5e+226)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(fma(Float64(Float64(180.0 / angle_m) / pi), Float64(Float64(Float64(angle_m * pi) * Float64(Float64(angle_m * pi) / Float64(angle_m * 0.005555555555555556))) / t_1), t_2) / Float64(180.0 * Float64(180.0 / angle_m))))));
	else
		tmp = Float64(Float64(a * a) * Float64(t_0 * fma(b, Float64(b / Float64(a * a)), -1.0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[(180.0 / angle$95$m), $MachinePrecision] + -180.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(32400.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * (-Pi)), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+25], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+133], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(N[(32400.0 * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] / N[(180.0 / N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+226], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(N[(180.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision] * N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] / N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] / N[(180.0 * N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(t$95$0 * N[(b * N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000018e25

   1. Initial program 56.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 75.8%

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

   if 2.00000000000000018e25 < (/.f64 angle #s(literal 180 binary64)) < 1e133

   1. Initial program 30.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 31.1%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. frac-add N/A

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

      13. /-lowering-/.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}}{180 \cdot \frac{180}{angle}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \color{blue}{\frac{180}{angle}}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, \color{blue}{180 \cdot \mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      20. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{180 \cdot \frac{180}{angle}}}\right)\right) \]

      22. /-lowering-/.f64 31.5

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

   6. Applied egg-rr 31.5%

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

      1. clear-num N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \color{blue}{\frac{1}{\frac{angle}{180}}}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      4. div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      6. metadata-eval 36.8

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

   8. Applied egg-rr 36.8%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle}\right), \frac{1}{\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot \mathsf{PI}\left(\right)}, \mathsf{neg}\left(\frac{\left(180 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(180 \cdot \mathsf{PI}\left(\right)\right)}{\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot \mathsf{PI}\left(\right)}\right)\right)}}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

   10. Applied egg-rr 54.5%

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)}{angle \cdot angle}, \frac{1}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, -\frac{\left(\pi \cdot \pi\right) \cdot 32400}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}\right)}}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right) \]

   if 1e133 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000005e226

   1. Initial program 29.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 33.1%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. frac-add N/A

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

      13. /-lowering-/.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}}{180 \cdot \frac{180}{angle}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \color{blue}{\frac{180}{angle}}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, \color{blue}{180 \cdot \mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      20. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{180 \cdot \frac{180}{angle}}}\right)\right) \]

      22. /-lowering-/.f64 39.2

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

   6. Applied egg-rr 39.2%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

   8. Applied egg-rr 54.2%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{180}{angle}}{\pi}, \frac{\left(\pi \cdot angle\right) \cdot \frac{\pi \cdot angle}{angle \cdot 0.005555555555555556}}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, -\frac{\left(\pi \cdot \pi\right) \cdot 32400}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}\right)}}{180 \cdot \frac{180}{angle}}\right)\right) \]

   if 5.0000000000000005e226 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 10.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(-1 \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\frac{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b + -1 \cdot b\right)}{a} + \frac{{b}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{a}^{2}}\right)\right)} \]

   7. Simplified 70.3%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+133}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)}{angle \cdot angle}, \frac{1}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, \frac{32400 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right) \cdot \left(-\pi\right)}\right)}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{\frac{180}{angle}}{\pi}, \frac{\left(angle \cdot \pi\right) \cdot \frac{angle \cdot \pi}{angle \cdot 0.005555555555555556}}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, \frac{32400 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right) \cdot \left(-\pi\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(angle\_m, \frac{180}{angle\_m}, -180\right)\\ t_1 := \frac{180}{angle\_m \cdot 0.005555555555555556}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(angle\_m \cdot \pi\right)\right)}{angle\_m \cdot angle\_m}, \frac{1}{\pi \cdot t\_0}, \frac{32400 \cdot \left(\pi \cdot \pi\right)}{t\_0 \cdot \left(-\pi\right)}\right)}{t\_1}\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+160}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\sqrt{\pi} \cdot \left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \sqrt{\pi}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - \frac{b \cdot a}{b}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(angle\_m \cdot \pi, \frac{180}{angle\_m}, 180 \cdot \pi\right)}{t\_1}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (fma angle_m (/ 180.0 angle_m) -180.0))
        (t_1 (/ 180.0 (* angle_m 0.005555555555555556))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+25)
      (* (+ b a) (* (- b a) (sin (* (* angle_m PI) 0.011111111111111112))))
      (if (<= (/ angle_m 180.0) 2e+133)
        (*
         (+ b a)
         (*
          (- b a)
          (sin
           (/
            (fma
             (/
              (* 32400.0 (* (* angle_m PI) (* angle_m PI)))
              (* angle_m angle_m))
             (/ 1.0 (* PI t_0))
             (/ (* 32400.0 (* PI PI)) (* t_0 (- PI))))
            t_1))))
        (if (<= (/ angle_m 180.0) 1e+160)
          (*
           (+ b a)
           (*
            (- b a)
            (sin
             (* (sqrt PI) (* (* angle_m 0.011111111111111112) (sqrt PI))))))
          (*
           (+ b a)
           (*
            (- b (/ (* b a) b))
            (sin
             (/
              (fma (* angle_m PI) (/ 180.0 angle_m) (* 180.0 PI))
              t_1))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = fma(angle_m, (180.0 / angle_m), -180.0);
	double t_1 = 180.0 / (angle_m * 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2e+25) {
		tmp = (b + a) * ((b - a) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)));
	} else if ((angle_m / 180.0) <= 2e+133) {
		tmp = (b + a) * ((b - a) * sin((fma(((32400.0 * ((angle_m * ((double) M_PI)) * (angle_m * ((double) M_PI)))) / (angle_m * angle_m)), (1.0 / (((double) M_PI) * t_0)), ((32400.0 * (((double) M_PI) * ((double) M_PI))) / (t_0 * -((double) M_PI)))) / t_1)));
	} else if ((angle_m / 180.0) <= 1e+160) {
		tmp = (b + a) * ((b - a) * sin((sqrt(((double) M_PI)) * ((angle_m * 0.011111111111111112) * sqrt(((double) M_PI))))));
	} else {
		tmp = (b + a) * ((b - ((b * a) / b)) * sin((fma((angle_m * ((double) M_PI)), (180.0 / angle_m), (180.0 * ((double) M_PI))) / t_1)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = fma(angle_m, Float64(180.0 / angle_m), -180.0)
	t_1 = Float64(180.0 / Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+25)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))));
	elseif (Float64(angle_m / 180.0) <= 2e+133)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(fma(Float64(Float64(32400.0 * Float64(Float64(angle_m * pi) * Float64(angle_m * pi))) / Float64(angle_m * angle_m)), Float64(1.0 / Float64(pi * t_0)), Float64(Float64(32400.0 * Float64(pi * pi)) / Float64(t_0 * Float64(-pi)))) / t_1))));
	elseif (Float64(angle_m / 180.0) <= 1e+160)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(sqrt(pi) * Float64(Float64(angle_m * 0.011111111111111112) * sqrt(pi))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - Float64(Float64(b * a) / b)) * sin(Float64(fma(Float64(angle_m * pi), Float64(180.0 / angle_m), Float64(180.0 * pi)) / t_1))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(180.0 / angle$95$m), $MachinePrecision] + -180.0), $MachinePrecision]}, Block[{t$95$1 = N[(180.0 / N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+25], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+133], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(N[(32400.0 * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(32400.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * (-Pi)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+160], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - N[(N[(b * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(180.0 / angle$95$m), $MachinePrecision] + N[(180.0 * Pi), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000018e25

   1. Initial program 56.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 75.8%

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

   if 2.00000000000000018e25 < (/.f64 angle #s(literal 180 binary64)) < 2e133

   1. Initial program 29.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 29.5%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. frac-add N/A

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

      13. /-lowering-/.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}}{180 \cdot \frac{180}{angle}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \color{blue}{\frac{180}{angle}}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, \color{blue}{180 \cdot \mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      20. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{180 \cdot \frac{180}{angle}}}\right)\right) \]

      22. /-lowering-/.f64 30.0

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

   6. Applied egg-rr 30.0%

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

      1. clear-num N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \color{blue}{\frac{1}{\frac{angle}{180}}}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      4. div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      6. metadata-eval 35.0

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

   8. Applied egg-rr 35.0%

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

      1. flip-+ N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle}\right), \frac{1}{\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot \mathsf{PI}\left(\right)}, \mathsf{neg}\left(\frac{\left(180 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(180 \cdot \mathsf{PI}\left(\right)\right)}{\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{180}{angle} - 180 \cdot \mathsf{PI}\left(\right)}\right)\right)}}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

   10. Applied egg-rr 56.8%

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)}{angle \cdot angle}, \frac{1}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, -\frac{\left(\pi \cdot \pi\right) \cdot 32400}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}\right)}}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right) \]

   if 2e133 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000001e160

   1. Initial program 44.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 40.1%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. add-sqr-sqrt N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{90} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{1}{90}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{1}{90}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \frac{1}{90}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

      13. PI-lowering-PI.f64 60.1

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

   6. Applied egg-rr 60.1%

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

   if 1.00000000000000001e160 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 19.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 43.9%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. frac-add N/A

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

      13. /-lowering-/.f64 N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}}{180 \cdot \frac{180}{angle}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot angle}, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \color{blue}{\frac{180}{angle}}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, \color{blue}{180 \cdot \mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      20. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{180 \cdot \frac{180}{angle}}\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{180 \cdot \frac{180}{angle}}}\right)\right) \]

      22. /-lowering-/.f64 45.1

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

   6. Applied egg-rr 45.1%

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

      1. clear-num N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{180 \cdot \color{blue}{\frac{1}{\frac{angle}{180}}}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\frac{180}{\frac{angle}{180}}}}\right)\right) \]

      4. div-inv N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{\color{blue}{angle \cdot \frac{1}{180}}}}\right)\right) \]

      6. metadata-eval 46.1

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

   8. Applied egg-rr 46.1%

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

   9. Taylor expanded in b around inf

      \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{\left(b \cdot \left(1 + -1 \cdot \frac{a}{b}\right)\right)} \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{\left(1 \cdot b + \left(-1 \cdot \frac{a}{b}\right) \cdot b\right)} \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       2. *-lft-identity N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(\color{blue}{b} + \left(-1 \cdot \frac{a}{b}\right) \cdot b\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b + \color{blue}{\left(\mathsf{neg}\left(\frac{a}{b}\right)\right)} \cdot b\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       4. distribute-lft-neg-out N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b + \color{blue}{\left(\mathsf{neg}\left(\frac{a}{b} \cdot b\right)\right)}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       5. unsub-neg N/A

          \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{\left(b - \frac{a}{b} \cdot b\right)} \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\color{blue}{\left(b - \frac{a}{b} \cdot b\right)} \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       7. associate-*l/ N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - \color{blue}{\frac{a \cdot b}{b}}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - \color{blue}{\frac{a \cdot b}{b}}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot angle, \frac{180}{angle}, 180 \cdot \mathsf{PI}\left(\right)\right)}{\frac{180}{angle \cdot \frac{1}{180}}}\right)\right) \]

       9. *-lowering-*.f64 46.1

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

   11. Simplified 46.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{\mathsf{fma}\left(\frac{32400 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)}{angle \cdot angle}, \frac{1}{\pi \cdot \mathsf{fma}\left(angle, \frac{180}{angle}, -180\right)}, \frac{32400 \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(angle, \frac{180}{angle}, -180\right) \cdot \left(-\pi\right)}\right)}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+160}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\sqrt{\pi} \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \sqrt{\pi}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - \frac{b \cdot a}{b}\right) \cdot \sin \left(\frac{\mathsf{fma}\left(angle \cdot \pi, \frac{180}{angle}, 180 \cdot \pi\right)}{\frac{180}{angle \cdot 0.005555555555555556}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-221}:\\ \;\;\;\;\left(a \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(b \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -4e-221)
    (* (* a (* angle_m PI)) (* a -0.011111111111111112))
    (* (* (* angle_m PI) 0.011111111111111112) (* b (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -4e-221) {
		tmp = (a * (angle_m * ((double) M_PI))) * (a * -0.011111111111111112);
	} else {
		tmp = ((angle_m * ((double) M_PI)) * 0.011111111111111112) * (b * (b - a));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -4e-221) {
		tmp = (a * (angle_m * Math.PI)) * (a * -0.011111111111111112);
	} else {
		tmp = ((angle_m * Math.PI) * 0.011111111111111112) * (b * (b - a));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -4e-221:
		tmp = (a * (angle_m * math.pi)) * (a * -0.011111111111111112)
	else:
		tmp = ((angle_m * math.pi) * 0.011111111111111112) * (b * (b - a))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -4e-221)
		tmp = Float64(Float64(a * Float64(angle_m * pi)) * Float64(a * -0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(angle_m * pi) * 0.011111111111111112) * Float64(b * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -4e-221)
		tmp = (a * (angle_m * pi)) * (a * -0.011111111111111112);
	else
		tmp = ((angle_m * pi) * 0.011111111111111112) * (b * (b - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -4e-221], N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.00000000000000007e-221

   1. Initial program 49.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 47.8

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

   5. Simplified 47.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 47.3

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

   8. Simplified 47.3%

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

      1. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right) \]

      7. PI-lowering-PI.f64 57.5

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

   10. Applied egg-rr 57.5%

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot \left(\frac{-1}{90} \cdot a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       7. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(a \cdot \frac{-1}{90}\right)} \]

       8. *-lowering-*.f64 57.6

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

   12. Applied egg-rr 57.6%

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

   if -4.00000000000000007e-221 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 53.5

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

   5. Simplified 53.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b} \cdot \left(b - a\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 53.2%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-221}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(b \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;\left(a \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.011111111111111112, angle\_m \cdot \pi, 0\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -4e-173)
    (* (* a (* angle_m PI)) (* a -0.011111111111111112))
    (* (* b b) (fma 0.011111111111111112 (* angle_m PI) 0.0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -4e-173) {
		tmp = (a * (angle_m * ((double) M_PI))) * (a * -0.011111111111111112);
	} else {
		tmp = (b * b) * fma(0.011111111111111112, (angle_m * ((double) M_PI)), 0.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -4e-173)
		tmp = Float64(Float64(a * Float64(angle_m * pi)) * Float64(a * -0.011111111111111112));
	else
		tmp = Float64(Float64(b * b) * fma(0.011111111111111112, Float64(angle_m * pi), 0.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -4e-173], N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.0000000000000002e-173

   1. Initial program 49.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 48.2

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

   5. Simplified 48.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 47.6

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

   8. Simplified 47.6%

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

      1. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right) \]

      7. PI-lowering-PI.f64 58.1

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

   10. Applied egg-rr 58.1%

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot \left(\frac{-1}{90} \cdot a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       7. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(a \cdot \frac{-1}{90}\right)} \]

       8. *-lowering-*.f64 58.2

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

   12. Applied egg-rr 58.2%

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

   if -4.0000000000000002e-173 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 51.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)}{b}\right)} \]

   8. Simplified 51.9%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;\left(a \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -4e-173)
    (* (* a (* angle_m PI)) (* a -0.011111111111111112))
    (* 0.011111111111111112 (* angle_m (* PI (* b b)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -4e-173) {
		tmp = (a * (angle_m * ((double) M_PI))) * (a * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * (b * b)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -4e-173) {
		tmp = (a * (angle_m * Math.PI)) * (a * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * (b * b)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -4e-173:
		tmp = (a * (angle_m * math.pi)) * (a * -0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * (b * b)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -4e-173)
		tmp = Float64(Float64(a * Float64(angle_m * pi)) * Float64(a * -0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b * b))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -4e-173)
		tmp = (a * (angle_m * pi)) * (a * -0.011111111111111112);
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * (b * b)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -4e-173], N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.0000000000000002e-173

   1. Initial program 49.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 48.2

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

   5. Simplified 48.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 47.6

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

   8. Simplified 47.6%

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

      1. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right) \]

      7. PI-lowering-PI.f64 58.1

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

   10. Applied egg-rr 58.1%

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

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)} \cdot \left(\frac{-1}{90} \cdot a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot angle\right)\right) \cdot \left(\frac{-1}{90} \cdot a\right) \]

       7. *-commutative N/A

          \[\leadsto \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \cdot \color{blue}{\left(a \cdot \frac{-1}{90}\right)} \]

       8. *-lowering-*.f64 58.2

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

   12. Applied egg-rr 58.2%

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

   if -4.0000000000000002e-173 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 51.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {b}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

      7. *-lowering-*.f64 51.9

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

   8. Simplified 51.9%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -4e-173)
    (* -0.011111111111111112 (* a (* a (* angle_m PI))))
    (* 0.011111111111111112 (* angle_m (* PI (* b b)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -4e-173) {
		tmp = -0.011111111111111112 * (a * (a * (angle_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * (b * b)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -4e-173) {
		tmp = -0.011111111111111112 * (a * (a * (angle_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * (b * b)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -4e-173:
		tmp = -0.011111111111111112 * (a * (a * (angle_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * (b * b)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -4e-173)
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(a * Float64(angle_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b * b))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -4e-173)
		tmp = -0.011111111111111112 * (a * (a * (angle_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * (b * b)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -4e-173], N[(-0.011111111111111112 * N[(a * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.0000000000000002e-173

   1. Initial program 49.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 48.2

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

   5. Simplified 48.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. PI-lowering-PI.f64 47.6

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

   8. Simplified 47.6%

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

      1. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{90} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right) \]

      7. PI-lowering-PI.f64 58.1

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

   10. Applied egg-rr 58.1%

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

   if -4.0000000000000002e-173 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 51.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {b}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

      7. *-lowering-*.f64 51.9

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

   8. Simplified 51.9%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+159}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+159)
    (*
     (+ b a)
     (*
      angle_m
      (fma
       (* (* angle_m angle_m) -2.2862368541380886e-7)
       (* (- b a) (* PI (* PI PI)))
       (* 0.011111111111111112 (* (- b a) PI)))))
    (* (sin (* (* angle_m PI) 0.011111111111111112)) (* (+ b a) (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+159) {
		tmp = (b + a) * (angle_m * fma(((angle_m * angle_m) * -2.2862368541380886e-7), ((b - a) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (0.011111111111111112 * ((b - a) * ((double) M_PI)))));
	} else {
		tmp = sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)) * ((b + a) * (b - a));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+159)
		tmp = Float64(Float64(b + a) * Float64(angle_m * fma(Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7), Float64(Float64(b - a) * Float64(pi * Float64(pi * pi))), Float64(0.011111111111111112 * Float64(Float64(b - a) * pi)))));
	else
		tmp = Float64(sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)) * Float64(Float64(b + a) * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+159], N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999993e158

   1. Initial program 54.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 71.0%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Simplified 69.2%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 9.9999999999999993e158 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 19.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. difference-of-squares N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. 2-sin N/A

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

      12. count-2 N/A

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

      13. sin-lowering-sin.f64 N/A

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

      14. associate-*r/ N/A

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

      15. div-inv N/A

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

      16. associate-*r/ N/A

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

      17. div-inv N/A

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

   4. Applied egg-rr 43.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+159}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{+236}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2.6e+236)
    (* (+ b a) (* (- b a) (sin (* (* angle_m PI) 0.011111111111111112))))
    (*
     (+ b a)
     (*
      (- b a)
      (*
       angle_m
       (fma
        -2.2862368541380886e-7
        (* (* angle_m angle_m) (* PI (* PI PI)))
        (* PI 0.011111111111111112))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.6e+236) {
		tmp = (b + a) * ((b - a) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = (b + a) * ((b - a) * (angle_m * fma(-2.2862368541380886e-7, ((angle_m * angle_m) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 2.6e+236)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 2.6e+236], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.6e236

   1. Initial program 50.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 68.9%

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

   if 2.6e236 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{{angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      15. PI-lowering-PI.f64 75.1

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Simplified 75.1%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{+236}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+236}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2.3e+236)
    (* (+ b a) (* (- b a) (sin (* angle_m (* PI 0.011111111111111112)))))
    (*
     (+ b a)
     (*
      (- b a)
      (*
       angle_m
       (fma
        -2.2862368541380886e-7
        (* (* angle_m angle_m) (* PI (* PI PI)))
        (* PI 0.011111111111111112))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.3e+236) {
		tmp = (b + a) * ((b - a) * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (b + a) * ((b - a) * (angle_m * fma(-2.2862368541380886e-7, ((angle_m * angle_m) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 2.3e+236)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 2.3e+236], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 2.3e236

   1. Initial program 50.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 68.9%

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

      1. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 68.5

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

   6. Applied egg-rr 68.5%

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

   if 2.3e236 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 58.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{{angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      15. PI-lowering-PI.f64 75.1

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Simplified 75.1%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+236}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-104}:\\ \;\;\;\;a \cdot \left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 4.9e-104)
    (* a (* (sin (* (* angle_m PI) 0.011111111111111112)) (- a)))
    (*
     (+ b a)
     (*
      angle_m
      (fma
       (* (* angle_m angle_m) -2.2862368541380886e-7)
       (* (- b a) (* PI (* PI PI)))
       (* 0.011111111111111112 (* (- b a) PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.9e-104) {
		tmp = a * (sin(((angle_m * ((double) M_PI)) * 0.011111111111111112)) * -a);
	} else {
		tmp = (b + a) * (angle_m * fma(((angle_m * angle_m) * -2.2862368541380886e-7), ((b - a) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (0.011111111111111112 * ((b - a) * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 4.9e-104)
		tmp = Float64(a * Float64(sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)) * Float64(-a)));
	else
		tmp = Float64(Float64(b + a) * Float64(angle_m * fma(Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7), Float64(Float64(b - a) * Float64(pi * Float64(pi * pi))), Float64(0.011111111111111112 * Float64(Float64(b - a) * pi)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 4.9e-104], N[(a * N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 4.9000000000000003e-104

   1. Initial program 52.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{2}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(-1 \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(-1 \cdot {a}^{2}\right)} \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot a}\right)\right) \]

      13. *-lowering-*.f64 42.2

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

   7. Simplified 42.2%

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

      1. distribute-lft-neg-in N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot a\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      4. add-cbrt-cube N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      5. add-sqr-sqrt N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      6. associate-*r* N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      7. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      8. unpow1/3 N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \left(\color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      9. *-commutative N/A

         \[\leadsto \left(\sin \left(\color{blue}{\left(\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot angle\right)} \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

   9. Applied egg-rr 46.8%

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

       1. associate-*r* N/A

          \[\leadsto \left(\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \left(\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

       4. PI-lowering-PI.f64 47.0

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

   11. Applied egg-rr 47.0%

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

   if 4.9000000000000003e-104 < b

   1. Initial program 46.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Simplified 66.1%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-104}:\\ \;\;\;\;a \cdot \left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-105}:\\ \;\;\;\;a \cdot \left(\left(-a\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 2.9e-105)
    (* a (* (- a) (sin (* angle_m (* PI 0.011111111111111112)))))
    (*
     (+ b a)
     (*
      angle_m
      (fma
       (* (* angle_m angle_m) -2.2862368541380886e-7)
       (* (- b a) (* PI (* PI PI)))
       (* 0.011111111111111112 (* (- b a) PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.9e-105) {
		tmp = a * (-a * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (b + a) * (angle_m * fma(((angle_m * angle_m) * -2.2862368541380886e-7), ((b - a) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (0.011111111111111112 * ((b - a) * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 2.9e-105)
		tmp = Float64(a * Float64(Float64(-a) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b + a) * Float64(angle_m * fma(Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7), Float64(Float64(b - a) * Float64(pi * Float64(pi * pi))), Float64(0.011111111111111112 * Float64(Float64(b - a) * pi)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 2.9e-105], N[(a * N[((-a) * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 2.90000000000000003e-105

   1. Initial program 52.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{2}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(-1 \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(-1 \cdot {a}^{2}\right)} \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot a}\right)\right) \]

      13. *-lowering-*.f64 42.2

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

   7. Simplified 42.2%

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

      1. distribute-lft-neg-in N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot a\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      4. add-cbrt-cube N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      5. add-sqr-sqrt N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      6. associate-*r* N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      7. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      8. unpow1/3 N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \left(\color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      9. *-commutative N/A

         \[\leadsto \left(\sin \left(\color{blue}{\left(\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot angle\right)} \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

   9. Applied egg-rr 46.8%

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

       1. *-commutative N/A

          \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

       2. associate-*r* N/A

          \[\leadsto \left(\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right) \cdot angle\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right) \cdot angle\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \left(\sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)} \cdot angle\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

       5. PI-lowering-PI.f64 47.1

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

   11. Applied egg-rr 47.1%

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

   if 2.90000000000000003e-105 < b

   1. Initial program 46.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Simplified 66.1%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-105}:\\ \;\;\;\;a \cdot \left(\left(-a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 10^{-103}:\\ \;\;\;\;a \cdot \left(\left(-a\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 1e-103)
    (* a (* (- a) (sin (* PI (* angle_m 0.011111111111111112)))))
    (*
     (+ b a)
     (*
      angle_m
      (fma
       (* (* angle_m angle_m) -2.2862368541380886e-7)
       (* (- b a) (* PI (* PI PI)))
       (* 0.011111111111111112 (* (- b a) PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1e-103) {
		tmp = a * (-a * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	} else {
		tmp = (b + a) * (angle_m * fma(((angle_m * angle_m) * -2.2862368541380886e-7), ((b - a) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (0.011111111111111112 * ((b - a) * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 1e-103)
		tmp = Float64(a * Float64(Float64(-a) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b + a) * Float64(angle_m * fma(Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7), Float64(Float64(b - a) * Float64(pi * Float64(pi * pi))), Float64(0.011111111111111112 * Float64(Float64(b - a) * pi)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 1e-103], N[(a * N[((-a) * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999958e-104

   1. Initial program 52.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{2}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(-1 \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(-1 \cdot {a}^{2}\right)} \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot a}\right)\right) \]

      13. *-lowering-*.f64 42.2

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

   7. Simplified 42.2%

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

      1. distribute-lft-neg-in N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot a\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      4. add-cbrt-cube N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      5. add-sqr-sqrt N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      6. associate-*r* N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      7. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      8. unpow1/3 N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \left(\color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      9. *-commutative N/A

         \[\leadsto \left(\sin \left(\color{blue}{\left(\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot angle\right)} \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

   9. Applied egg-rr 46.8%

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

   if 9.99999999999999958e-104 < b

   1. Initial program 46.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      16. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   7. Simplified 66.1%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-103}:\\ \;\;\;\;a \cdot \left(\left(-a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;a \cdot \left(angle\_m \cdot \left(a \cdot \mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e+73)
    (* (+ b a) (* 0.011111111111111112 (* angle_m (* (- b a) PI))))
    (if (<= (/ angle_m 180.0) 2e+218)
      (*
       a
       (*
        angle_m
        (*
         a
         (fma
          (* angle_m (* angle_m (* PI (* PI PI))))
          2.2862368541380886e-7
          (* PI -0.011111111111111112)))))
      (* (* (* angle_m PI) 0.011111111111111112) (* a (- (- a) b)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+73) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * ((b - a) * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 2e+218) {
		tmp = a * (angle_m * (a * fma((angle_m * (angle_m * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))), 2.2862368541380886e-7, (((double) M_PI) * -0.011111111111111112))));
	} else {
		tmp = ((angle_m * ((double) M_PI)) * 0.011111111111111112) * (a * (-a - b));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+73)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a) * pi))));
	elseif (Float64(angle_m / 180.0) <= 2e+218)
		tmp = Float64(a * Float64(angle_m * Float64(a * fma(Float64(angle_m * Float64(angle_m * Float64(pi * Float64(pi * pi)))), 2.2862368541380886e-7, Float64(pi * -0.011111111111111112)))));
	else
		tmp = Float64(Float64(Float64(angle_m * pi) * 0.011111111111111112) * Float64(a * Float64(Float64(-a) - b)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+73], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+218], N[(a * N[(angle$95$m * N[(a * N[(N[(angle$95$m * N[(angle$95$m * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.2862368541380886e-7 + N[(Pi * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(a * N[((-a) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999993e73

   1. Initial program 55.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      5. --lowering--.f64 69.1

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

   7. Simplified 69.1%

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

   if 3.99999999999999993e73 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000017e218

   1. Initial program 32.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 35.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{2}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(-1 \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(-1 \cdot {a}^{2}\right)} \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot a}\right)\right) \]

      13. *-lowering-*.f64 22.9

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

   7. Simplified 22.9%

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

      1. distribute-lft-neg-in N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot a\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a} \]

      3. *-commutative N/A

         \[\leadsto \left(\sin \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      4. add-cbrt-cube N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      5. add-sqr-sqrt N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      6. associate-*r* N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \sqrt[3]{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      7. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \color{blue}{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      8. unpow1/3 N/A

         \[\leadsto \left(\sin \left(\left(angle \cdot \left(\color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

      9. *-commutative N/A

         \[\leadsto \left(\sin \left(\color{blue}{\left(\left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot angle\right)} \cdot \frac{1}{90}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right) \cdot a \]

   9. Applied egg-rr 25.9%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \cdot a \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \left(a \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right)\right)\right) \cdot a \]

       2. associate-*r* N/A

          \[\leadsto \left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right) \cdot a \]

       3. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\left(\frac{1}{4374000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}}\right)\right) \cdot a \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{4374000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)} \cdot a \]

       5. +-commutative N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(\left(\frac{1}{4374000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot a \]

       6. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\frac{1}{4374000} \cdot \left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

       7. associate-*r* N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{4374000} \cdot \color{blue}{\left(a \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

       8. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\frac{1}{4374000} \cdot \left(a \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

       9. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\color{blue}{\left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \frac{1}{4374000}} + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot a \]

   12. Simplified 29.6%

       \[\leadsto \color{blue}{\left(angle \cdot \left(a \cdot \mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\right)} \cdot a \]

   if 2.00000000000000017e218 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 42.3

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

   5. Simplified 42.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

      2. neg-lowering-neg.f64 50.9

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

   8. Simplified 50.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;a \cdot \left(angle \cdot \left(a \cdot \mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 10^{+215}:\\ \;\;\;\;angle\_m \cdot \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\pi \cdot \left(angle\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e+73)
    (* (+ b a) (* 0.011111111111111112 (* angle_m (* (- b a) PI))))
    (if (<= (/ angle_m 180.0) 1e+215)
      (*
       angle_m
       (*
        (* a a)
        (fma
         (* PI (* angle_m (* angle_m (* PI PI))))
         2.2862368541380886e-7
         (* PI -0.011111111111111112))))
      (* (* (* angle_m PI) 0.011111111111111112) (* a (- (- a) b)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+73) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * ((b - a) * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 1e+215) {
		tmp = angle_m * ((a * a) * fma((((double) M_PI) * (angle_m * (angle_m * (((double) M_PI) * ((double) M_PI))))), 2.2862368541380886e-7, (((double) M_PI) * -0.011111111111111112)));
	} else {
		tmp = ((angle_m * ((double) M_PI)) * 0.011111111111111112) * (a * (-a - b));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+73)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a) * pi))));
	elseif (Float64(angle_m / 180.0) <= 1e+215)
		tmp = Float64(angle_m * Float64(Float64(a * a) * fma(Float64(pi * Float64(angle_m * Float64(angle_m * Float64(pi * pi)))), 2.2862368541380886e-7, Float64(pi * -0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(angle_m * pi) * 0.011111111111111112) * Float64(a * Float64(Float64(-a) - b)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+73], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+215], N[(angle$95$m * N[(N[(a * a), $MachinePrecision] * N[(N[(Pi * N[(angle$95$m * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.2862368541380886e-7 + N[(Pi * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(a * N[((-a) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999993e73

   1. Initial program 55.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      5. --lowering--.f64 69.1

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

   7. Simplified 69.1%

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

   if 3.99999999999999993e73 < (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999907e214

   1. Initial program 32.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 35.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{2}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(-1 \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(-1 \cdot {a}^{2}\right)} \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(-1 \cdot {a}^{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(-1 \cdot {a}^{2}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left({a}^{2}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot a}\right)\right) \]

      13. *-lowering-*.f64 22.9

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

   7. Simplified 22.9%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto angle \cdot \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4374000} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right) \]

      3. associate-*l* N/A

         \[\leadsto angle \cdot \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\left(\frac{1}{4374000} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{4374000} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)} \]

      5. +-commutative N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(\frac{1}{4374000} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2} + \frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto angle \cdot \left(\color{blue}{\frac{1}{4374000} \cdot \left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)} + \frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto angle \cdot \left(\frac{1}{4374000} \cdot \color{blue}{\left({a}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)} + \frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto angle \cdot \left(\frac{1}{4374000} \cdot \left({a}^{2} \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) + \frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \frac{1}{4374000}} + \frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. Simplified 35.5%

       \[\leadsto \color{blue}{angle \cdot \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\pi \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)} \]

   if 9.99999999999999907e214 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 42.3

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

   5. Simplified 42.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

      2. neg-lowering-neg.f64 50.9

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

   8. Simplified 50.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+215}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\pi \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right), 2.2862368541380886 \cdot 10^{-7}, \pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b + a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (+ b a)
   (*
    angle_m
    (fma
     (* (* angle_m angle_m) -2.2862368541380886e-7)
     (* (- b a) (* PI (* PI PI)))
     (* 0.011111111111111112 (* (- b a) PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b + a) * (angle_m * fma(((angle_m * angle_m) * -2.2862368541380886e-7), ((b - a) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (0.011111111111111112 * ((b - a) * ((double) M_PI))))));
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(b + a) * Float64(angle_m * fma(Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7), Float64(Float64(b - a) * Float64(pi * Float64(pi * pi))), Float64(0.011111111111111112 * Float64(Float64(b - a) * pi))))))
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. --lowering--.f64 N/A

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

   12. 2-sin N/A

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

   13. count-2 N/A

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

4. Applied egg-rr 68.4%

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

5. Taylor expanded in angle around 0

   \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   10. cube-mult N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   13. PI-lowering-PI.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   16. PI-lowering-PI.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   17. PI-lowering-PI.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

7. Simplified 66.0%

   \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

8. Final simplification 66.0%

   \[\leadsto \left(b + a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right) \]
9. Add Preprocessing

Alternative 19: 62.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;t\_0 \cdot \left(b \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.011111111111111112)))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 2e+128)
      (* (+ b a) (* 0.011111111111111112 (* angle_m (* (- b a) PI))))
      (if (<= (/ angle_m 180.0) 2e+218)
        (* t_0 (* b (+ b a)))
        (* t_0 (* a (- (- a) b))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.011111111111111112;
	double tmp;
	if ((angle_m / 180.0) <= 2e+128) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * ((b - a) * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 2e+218) {
		tmp = t_0 * (b * (b + a));
	} else {
		tmp = t_0 * (a * (-a - b));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * Math.PI) * 0.011111111111111112;
	double tmp;
	if ((angle_m / 180.0) <= 2e+128) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * ((b - a) * Math.PI)));
	} else if ((angle_m / 180.0) <= 2e+218) {
		tmp = t_0 * (b * (b + a));
	} else {
		tmp = t_0 * (a * (-a - b));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (angle_m * math.pi) * 0.011111111111111112
	tmp = 0
	if (angle_m / 180.0) <= 2e+128:
		tmp = (b + a) * (0.011111111111111112 * (angle_m * ((b - a) * math.pi)))
	elif (angle_m / 180.0) <= 2e+218:
		tmp = t_0 * (b * (b + a))
	else:
		tmp = t_0 * (a * (-a - b))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.011111111111111112)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+128)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a) * pi))));
	elseif (Float64(angle_m / 180.0) <= 2e+218)
		tmp = Float64(t_0 * Float64(b * Float64(b + a)));
	else
		tmp = Float64(t_0 * Float64(a * Float64(Float64(-a) - b)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (angle_m * pi) * 0.011111111111111112;
	tmp = 0.0;
	if ((angle_m / 180.0) <= 2e+128)
		tmp = (b + a) * (0.011111111111111112 * (angle_m * ((b - a) * pi)));
	elseif ((angle_m / 180.0) <= 2e+218)
		tmp = t_0 * (b * (b + a));
	else
		tmp = t_0 * (a * (-a - b));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+128], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+218], N[(t$95$0 * N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(a * N[((-a) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.0000000000000002e128

   1. Initial program 55.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. 2-sin N/A

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

      13. count-2 N/A

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

   4. Applied egg-rr 72.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      5. --lowering--.f64 67.6

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

   7. Simplified 67.6%

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

   if 2.0000000000000002e128 < (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000017e218

   1. Initial program 24.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 16.3

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

   5. Simplified 16.3%

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

   6. Taylor expanded in b around inf

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{b}\right) \]
   7. Step-by-step derivation

   8. Simplified 21.3%

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

   if 2.00000000000000017e218 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

      12. --lowering--.f64 42.3

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

   5. Simplified 42.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(-1 \cdot a\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) \]

      2. neg-lowering-neg.f64 50.9

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

   8. Simplified 50.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+128}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+218}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(b \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 61.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (+ b a)
   (*
    (- b a)
    (*
     angle_m
     (fma
      -2.2862368541380886e-7
      (* (* angle_m angle_m) (* PI (* PI PI)))
      (* PI 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b + a) * ((b - a) * (angle_m * fma(-2.2862368541380886e-7, ((angle_m * angle_m) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (((double) M_PI) * 0.011111111111111112)))));
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112))))))
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. --lowering--.f64 N/A

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

   12. 2-sin N/A

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

   13. count-2 N/A

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

4. Applied egg-rr 68.4%

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

5. Taylor expanded in angle around 0

   \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{{angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \color{blue}{\left(angle \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. cube-mult N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   12. PI-lowering-PI.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   13. PI-lowering-PI.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   15. PI-lowering-PI.f64 66.0

       \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

7. Simplified 66.0%

   \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]

8. Final simplification 66.0%

   \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right) \]
9. Add Preprocessing

Alternative 21: 37.5% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   9. difference-of-squares N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   12. --lowering--.f64 51.3

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

5. Simplified 51.3%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

   7. PI-lowering-PI.f64 34.2

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

8. Simplified 34.2%

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

   1. associate-*l* N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left(a \cdot \left(a \cdot angle\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \left(\color{blue}{\left(a \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. PI-lowering-PI.f64 38.0

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

10. Applied egg-rr 38.0%

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

11. Final simplification 38.0%

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

Alternative 22: 37.5% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

   9. difference-of-squares N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   12. --lowering--.f64 51.3

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

5. Simplified 51.3%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \]

   7. PI-lowering-PI.f64 34.2

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

8. Simplified 34.2%

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

   1. associate-*l* N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{-1}{90} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

   3. associate-*l* N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{-1}{90} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right) \]

   7. PI-lowering-PI.f64 38.0

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

10. Applied egg-rr 38.0%

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

11. Final simplification 38.0%

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

Reproduce

herbie shell --seed 2024198 
(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)))))