ab-angle->ABCF B

Percentage Accurate: 54.4% → 66.0%

Time: 15.2s

Alternatives: 13

Speedup: N/A×

• 32.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.0% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\\ t_2 := \sin \left(\frac{t\_1 + t\_1}{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\left(2 \cdot \left(t\_2 \cdot \cos \left(\frac{t\_1 - 0.5 \cdot \pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(t\_2 \cdot \cos \left(\frac{0}{2}\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, a + b, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle\_m \cdot angle\_m\right), {\pi}^{5} \cdot \left(a + b\right), \left(-2.8577960676726107 \cdot 10^{-8} \cdot {\pi}^{3}\right) \cdot \left(a + b\right)\right) \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot angle\_m\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* PI (/ angle_m 180.0)))
        (t_1 (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0)))
        (t_2 (sin (/ (+ t_1 t_1) 2.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         4e+298)
      (*
       (* 2.0 (* t_2 (cos (/ (- t_1 (* 0.5 PI)) 2.0))))
       (* (* (sin (* (* angle_m PI) 0.005555555555555556)) (+ a b)) (- b a)))
      (*
       (* 2.0 (* t_2 (cos (/ 0.0 2.0))))
       (*
        (*
         (fma
          (* 0.005555555555555556 PI)
          (+ a b)
          (*
           (fma
            (* 4.410179116778721e-14 (* angle_m angle_m))
            (* (pow PI 5.0) (+ a b))
            (* (* -2.8577960676726107e-8 (pow PI 3.0)) (+ a b)))
           (* angle_m angle_m)))
         angle_m)
        (- 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 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0));
	double t_2 = sin(((t_1 + t_1) / 2.0));
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 4e+298) {
		tmp = (2.0 * (t_2 * cos(((t_1 - (0.5 * ((double) M_PI))) / 2.0)))) * ((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * (a + b)) * (b - a));
	} else {
		tmp = (2.0 * (t_2 * cos((0.0 / 2.0)))) * ((fma((0.005555555555555556 * ((double) M_PI)), (a + b), (fma((4.410179116778721e-14 * (angle_m * angle_m)), (pow(((double) M_PI), 5.0) * (a + b)), ((-2.8577960676726107e-8 * pow(((double) M_PI), 3.0)) * (a + b))) * (angle_m * angle_m))) * angle_m) * (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(pi * Float64(angle_m / 180.0))
	t_1 = fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0))
	t_2 = sin(Float64(Float64(t_1 + t_1) / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 4e+298)
		tmp = Float64(Float64(2.0 * Float64(t_2 * cos(Float64(Float64(t_1 - Float64(0.5 * pi)) / 2.0)))) * Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * Float64(a + b)) * Float64(b - a)));
	else
		tmp = Float64(Float64(2.0 * Float64(t_2 * cos(Float64(0.0 / 2.0)))) * Float64(Float64(fma(Float64(0.005555555555555556 * pi), Float64(a + b), Float64(fma(Float64(4.410179116778721e-14 * Float64(angle_m * angle_m)), Float64((pi ^ 5.0) * Float64(a + b)), Float64(Float64(-2.8577960676726107e-8 * (pi ^ 3.0)) * Float64(a + b))) * Float64(angle_m * angle_m))) * angle_m) * 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_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(t$95$1 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $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], 4e+298], N[(N[(2.0 * N[(t$95$2 * N[Cos[N[(N[(t$95$1 - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$2 * N[Cos[N[(0.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(a + b), $MachinePrecision] + N[(N[(N[(4.410179116778721e-14 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.8577960676726107e-8 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $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))))) < 3.9999999999999998e298

   1. Initial program 55.6%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 62.8%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \frac{1}{2} \cdot \mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \frac{1}{2} \cdot \mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-PI.f64 62.9

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - 0.5 \cdot \pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   13. Applied rewrites 62.9%

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - 0.5 \cdot \pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if 3.9999999999999998e298 < (*.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 45.1%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 78.1%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(\left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(\left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \]

   13. Applied rewrites 81.6%

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, a + b, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle \cdot angle\right), {\pi}^{5} \cdot \left(a + b\right), \left(-2.8577960676726107 \cdot 10^{-8} \cdot {\pi}^{3}\right) \cdot \left(a + b\right)\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 4 \cdot 10^{+298}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - 0.5 \cdot \pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{0}{2}\right)\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, a + b, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle \cdot angle\right), {\pi}^{5} \cdot \left(a + b\right), \left(-2.8577960676726107 \cdot 10^{-8} \cdot {\pi}^{3}\right) \cdot \left(a + b\right)\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.9% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_2 := \left(\sin t\_1 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\\ t_3 := \mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(\frac{t\_3 + t\_3}{2}\right) \cdot \cos \left(\frac{t\_3 - 0.5 \cdot \pi}{2}\right)\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos t\_1\right) \cdot t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (* (* angle_m PI) 0.005555555555555556))
        (t_2 (* (* (sin t_1) (+ a b)) (- b a)))
        (t_3 (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -2e+239)
      (*
       (* 2.0 (* (sin (/ (+ t_3 t_3) 2.0)) (cos (/ (- t_3 (* 0.5 PI)) 2.0))))
       t_2)
      (* (* 2.0 (cos t_1)) t_2)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_2 = (sin(t_1) * (a + b)) * (b - a);
	double t_3 = fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0));
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+239) {
		tmp = (2.0 * (sin(((t_3 + t_3) / 2.0)) * cos(((t_3 - (0.5 * ((double) M_PI))) / 2.0)))) * t_2;
	} else {
		tmp = (2.0 * cos(t_1)) * t_2;
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_2 = Float64(Float64(sin(t_1) * Float64(a + b)) * Float64(b - a))
	t_3 = fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+239)
		tmp = Float64(Float64(2.0 * Float64(sin(Float64(Float64(t_3 + t_3) / 2.0)) * cos(Float64(Float64(t_3 - Float64(0.5 * pi)) / 2.0)))) * t_2);
	else
		tmp = Float64(Float64(2.0 * cos(t_1)) * t_2);
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $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], -2e+239], N[(N[(2.0 * N[(N[Sin[N[(N[(t$95$3 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$3 - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(2.0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$2), $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))))) < -1.99999999999999998e239

   1. Initial program 48.8%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 73.5%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \frac{1}{2} \cdot \mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \frac{1}{2} \cdot \mathsf{PI}\left(\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-PI.f64 78.7

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - 0.5 \cdot \pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   13. Applied rewrites 78.7%

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - 0.5 \cdot \pi}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if -1.99999999999999998e239 < (*.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 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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 63.0%

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

Alternative 3: 66.8% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_2 := \left(\sin t\_1 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right) + 0.5 \cdot \pi}{2}\right) \cdot \cos \left(\frac{0}{2}\right)\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos t\_1\right) \cdot t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (* (* angle_m PI) 0.005555555555555556))
        (t_2 (* (* (sin t_1) (+ a b)) (- b a))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -2e+239)
      (*
       (*
        2.0
        (*
         (sin
          (/
           (+ (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0)) (* 0.5 PI))
           2.0))
         (cos (/ 0.0 2.0))))
       t_2)
      (* (* 2.0 (cos t_1)) t_2)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_2 = (sin(t_1) * (a + b)) * (b - a);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+239) {
		tmp = (2.0 * (sin(((fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)) + (0.5 * ((double) M_PI))) / 2.0)) * cos((0.0 / 2.0)))) * t_2;
	} else {
		tmp = (2.0 * cos(t_1)) * t_2;
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_2 = Float64(Float64(sin(t_1) * Float64(a + b)) * Float64(b - a))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+239)
		tmp = Float64(Float64(2.0 * Float64(sin(Float64(Float64(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)) + Float64(0.5 * pi)) / 2.0)) * cos(Float64(0.0 / 2.0)))) * t_2);
	else
		tmp = Float64(Float64(2.0 * cos(t_1)) * t_2);
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $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], -2e+239], N[(N[(2.0 * N[(N[Sin[N[(N[(N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(2.0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$2), $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))))) < -1.99999999999999998e239

   1. Initial program 48.8%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 73.5%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

       2. lift-PI.f64 75.5

          \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + 0.5 \cdot \pi}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   13. Applied rewrites 75.5%

       \[\leadsto \left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + 0.5 \cdot \pi}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) - \mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   if -1.99999999999999998e239 < (*.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 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 inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 63.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -2 \cdot 10^{+239}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right) + 0.5 \cdot \pi}{2}\right) \cdot \cos \left(\frac{0}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_2 := \left(\sin t\_1 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -2 \cdot 10^{+291}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \left(angle\_m \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot \cos \left(\frac{t\_1 + t\_1}{2}\right)\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos t\_1\right) \cdot t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (* (* angle_m PI) 0.005555555555555556))
        (t_2 (* (* (sin t_1) (+ a b)) (- b a))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -2e+291)
      (*
       (*
        2.0
        (*
         (cos
          (/
           (fma
            (* 0.005555555555555556 angle_m)
            PI
            (* (* angle_m PI) (- 0.005555555555555556)))
           2.0))
         (cos (/ (+ t_1 t_1) 2.0))))
       t_2)
      (* (* 2.0 (cos t_1)) t_2)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_2 = (sin(t_1) * (a + b)) * (b - a);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+291) {
		tmp = (2.0 * (cos((fma((0.005555555555555556 * angle_m), ((double) M_PI), ((angle_m * ((double) M_PI)) * -0.005555555555555556)) / 2.0)) * cos(((t_1 + t_1) / 2.0)))) * t_2;
	} else {
		tmp = (2.0 * cos(t_1)) * t_2;
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_2 = Float64(Float64(sin(t_1) * Float64(a + b)) * Float64(b - a))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+291)
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(Float64(angle_m * pi) * Float64(-0.005555555555555556))) / 2.0)) * cos(Float64(Float64(t_1 + t_1) / 2.0)))) * t_2);
	else
		tmp = Float64(Float64(2.0 * cos(t_1)) * t_2);
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $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], -2e+291], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(N[(angle$95$m * Pi), $MachinePrecision] * (-0.005555555555555556)), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$1 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(2.0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$2), $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))))) < -1.9999999999999999e291

   1. Initial program 52.3%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. cos-neg-rev N/A

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

      9. sum-cos N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 78.7%

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

   if -1.9999999999999999e291 < (*.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 53.6%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 61.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -2 \cdot 10^{+291}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot \left(-0.005555555555555556\right)\right)}{2}\right) \cdot \cos \left(\frac{\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556}{2}\right)\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_2 := 2 \cdot \cos t\_1\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -5 \cdot 10^{+294}:\\ \;\;\;\;t\_2 \cdot \left(\left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, a + b, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle\_m \cdot angle\_m\right), {\pi}^{5} \cdot \left(a + b\right), \left(-2.8577960676726107 \cdot 10^{-8} \cdot {\pi}^{3}\right) \cdot \left(a + b\right)\right) \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot angle\_m\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\left(\sin t\_1 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* PI (/ angle_m 180.0)))
        (t_1 (* (* angle_m PI) 0.005555555555555556))
        (t_2 (* 2.0 (cos t_1))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -5e+294)
      (*
       t_2
       (*
        (*
         (fma
          (* 0.005555555555555556 PI)
          (+ a b)
          (*
           (fma
            (* 4.410179116778721e-14 (* angle_m angle_m))
            (* (pow PI 5.0) (+ a b))
            (* (* -2.8577960676726107e-8 (pow PI 3.0)) (+ a b)))
           (* angle_m angle_m)))
         angle_m)
        (- b a)))
      (* t_2 (* (* (sin t_1) (+ a b)) (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_2 = 2.0 * cos(t_1);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -5e+294) {
		tmp = t_2 * ((fma((0.005555555555555556 * ((double) M_PI)), (a + b), (fma((4.410179116778721e-14 * (angle_m * angle_m)), (pow(((double) M_PI), 5.0) * (a + b)), ((-2.8577960676726107e-8 * pow(((double) M_PI), 3.0)) * (a + b))) * (angle_m * angle_m))) * angle_m) * (b - a));
	} else {
		tmp = t_2 * ((sin(t_1) * (a + b)) * (b - a));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_2 = Float64(2.0 * cos(t_1))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -5e+294)
		tmp = Float64(t_2 * Float64(Float64(fma(Float64(0.005555555555555556 * pi), Float64(a + b), Float64(fma(Float64(4.410179116778721e-14 * Float64(angle_m * angle_m)), Float64((pi ^ 5.0) * Float64(a + b)), Float64(Float64(-2.8577960676726107e-8 * (pi ^ 3.0)) * Float64(a + b))) * Float64(angle_m * angle_m))) * angle_m) * Float64(b - a)));
	else
		tmp = Float64(t_2 * Float64(Float64(sin(t_1) * Float64(a + b)) * 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_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Cos[t$95$1], $MachinePrecision]), $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], -5e+294], N[(t$95$2 * N[(N[(N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(a + b), $MachinePrecision] + N[(N[(N[(4.410179116778721e-14 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.8577960676726107e-8 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(a + b), $MachinePrecision]), $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))))) < -4.9999999999999999e294

   1. Initial program 53.3%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 80.6%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 81.8%

       \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, a + b, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle \cdot angle\right), {\pi}^{5} \cdot \left(a + b\right), \left(-2.8577960676726107 \cdot 10^{-8} \cdot {\pi}^{3}\right) \cdot \left(a + b\right)\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right) \cdot \left(\color{blue}{b} - a\right)\right) \]

   if -4.9999999999999999e294 < (*.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 53.4%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 61.2%

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

Alternative 6: 67.0% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_1 := 2 \cdot \cos t\_0\\ t_2 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{+236}:\\ \;\;\;\;t\_1 \cdot \left(\left(\mathsf{fma}\left(b, \frac{t\_2}{a}, t\_2\right) \cdot a\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, a + b, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle\_m \cdot angle\_m\right), {\pi}^{5} \cdot \left(a + b\right), \left(-2.8577960676726107 \cdot 10^{-8} \cdot {\pi}^{3}\right) \cdot \left(a + b\right)\right) \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot angle\_m\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

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 PI) 0.005555555555555556))
        (t_1 (* 2.0 (cos t_0)))
        (t_2 (sin t_0)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e+236)
      (* t_1 (* (* (fma b (/ t_2 a) t_2) a) (- b a)))
      (*
       t_1
       (*
        (*
         (fma
          (* 0.005555555555555556 PI)
          (+ a b)
          (*
           (fma
            (* 4.410179116778721e-14 (* angle_m angle_m))
            (* (pow PI 5.0) (+ a b))
            (* (* -2.8577960676726107e-8 (pow PI 3.0)) (+ a b)))
           (* angle_m angle_m)))
         angle_m)
        (- 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 * ((double) M_PI)) * 0.005555555555555556;
	double t_1 = 2.0 * cos(t_0);
	double t_2 = sin(t_0);
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e+236) {
		tmp = t_1 * ((fma(b, (t_2 / a), t_2) * a) * (b - a));
	} else {
		tmp = t_1 * ((fma((0.005555555555555556 * ((double) M_PI)), (a + b), (fma((4.410179116778721e-14 * (angle_m * angle_m)), (pow(((double) M_PI), 5.0) * (a + b)), ((-2.8577960676726107e-8 * pow(((double) M_PI), 3.0)) * (a + b))) * (angle_m * angle_m))) * angle_m) * (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 * pi) * 0.005555555555555556)
	t_1 = Float64(2.0 * cos(t_0))
	t_2 = sin(t_0)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e+236)
		tmp = Float64(t_1 * Float64(Float64(fma(b, Float64(t_2 / a), t_2) * a) * Float64(b - a)));
	else
		tmp = Float64(t_1 * Float64(Float64(fma(Float64(0.005555555555555556 * pi), Float64(a + b), Float64(fma(Float64(4.410179116778721e-14 * Float64(angle_m * angle_m)), Float64((pi ^ 5.0) * Float64(a + b)), Float64(Float64(-2.8577960676726107e-8 * (pi ^ 3.0)) * Float64(a + b))) * Float64(angle_m * angle_m))) * angle_m) * 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_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+236], N[(t$95$1 * N[(N[(N[(b * N[(t$95$2 / a), $MachinePrecision] + t$95$2), $MachinePrecision] * a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * N[(a + b), $MachinePrecision] + N[(N[(N[(4.410179116778721e-14 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.8577960676726107e-8 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000000000005e236

   1. Initial program 53.8%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 62.7%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 62.5%

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

   if 1.00000000000000005e236 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 52.0%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(angle \cdot \left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(\left(\left(\frac{1}{180} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{1}{22674816000000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right)\right) \]

   11. Applied rewrites 79.8%

       \[\leadsto \left(2 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, a + b, \mathsf{fma}\left(4.410179116778721 \cdot 10^{-14} \cdot \left(angle \cdot angle\right), {\pi}^{5} \cdot \left(a + b\right), \left(-2.8577960676726107 \cdot 10^{-8} \cdot {\pi}^{3}\right) \cdot \left(a + b\right)\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right) \cdot \left(\color{blue}{b} - a\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.1% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_1 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(\mathsf{fma}\left(b, \frac{t\_1}{a}, t\_1\right) \cdot a\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot 1\right) \cdot \left(\left(t\_1 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

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 PI) 0.005555555555555556)) (t_1 (sin t_0)))
   (*
    angle_s
    (if (<= (pow b 2.0) 5e+153)
      (* (* 2.0 (cos t_0)) (* (* (fma b (/ t_1 a) t_1) a) (- b a)))
      (* (* 2.0 1.0) (* (* t_1 (+ a b)) (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_1 = sin(t_0);
	double tmp;
	if (pow(b, 2.0) <= 5e+153) {
		tmp = (2.0 * cos(t_0)) * ((fma(b, (t_1 / a), t_1) * a) * (b - a));
	} else {
		tmp = (2.0 * 1.0) * ((t_1 * (a + b)) * (b - a));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_1 = sin(t_0)
	tmp = 0.0
	if ((b ^ 2.0) <= 5e+153)
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(fma(b, Float64(t_1 / a), t_1) * a) * Float64(b - a)));
	else
		tmp = Float64(Float64(2.0 * 1.0) * Float64(Float64(t_1 * Float64(a + b)) * 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_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e+153], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * N[(t$95$1 / a), $MachinePrecision] + t$95$1), $MachinePrecision] * a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * 1.0), $MachinePrecision] * N[(N[(t$95$1 * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 5.00000000000000018e153

   1. Initial program 57.1%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 65.7%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 65.7%

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

   if 5.00000000000000018e153 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 47.0%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 64.7%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 66.6%

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

Alternative 8: 66.3% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(2 \cdot 1\right) \cdot \left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* 2.0 1.0)
   (* (* (sin (* (* angle_m PI) 0.005555555555555556)) (+ a b)) (- b a)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((2.0 * 1.0) * ((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * (a + b)) * (b - a)));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((2.0 * 1.0) * ((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * (a + b)) * (b - a)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((2.0 * 1.0) * ((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * (a + b)) * (b - a)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(2.0 * 1.0) * Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * Float64(a + b)) * Float64(b - a))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((2.0 * 1.0) * ((sin(((angle_m * pi) * 0.005555555555555556)) * (a + b)) * (b - a)));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in angle around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. difference-of-squares N/A

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

   14. lower-*.f64 N/A

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

   15. lower-+.f64 N/A

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

   16. lower--.f64 55.2

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

5. Applied rewrites 55.2%

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

6. Taylor expanded in angle around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 65.4%

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

9. Taylor expanded in angle around 0

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

11. Applied rewrites 63.7%

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

Alternative 9: 56.3% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-313}:\\ \;\;\;\;\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle\_m \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-313)
      (* (* -2.0 (* a a)) (* (sin t_0) (cos t_0)))
      (*
       (* (* b b) 2.0)
       (*
        (sin (* (fma 0.011111111111111112 (* angle_m PI) PI) 0.5))
        (sin (* (* angle_m PI) 0.005555555555555556))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-313) {
		tmp = (-2.0 * (a * a)) * (sin(t_0) * cos(t_0));
	} else {
		tmp = ((b * b) * 2.0) * (sin((fma(0.011111111111111112, (angle_m * ((double) M_PI)), ((double) M_PI)) * 0.5)) * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-313)
		tmp = Float64(Float64(-2.0 * Float64(a * a)) * Float64(sin(t_0) * cos(t_0)));
	else
		tmp = Float64(Float64(Float64(b * b) * 2.0) * Float64(sin(Float64(fma(0.011111111111111112, Float64(angle_m * pi), pi) * 0.5)) * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-313], N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision] + Pi), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000001e-313

   1. Initial program 55.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \]

   5. Applied rewrites 57.0%

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

   if 1.00000000001e-313 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 51.2%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 65.4%

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

   11. Taylor expanded in a around 0

       \[\leadsto 2 \cdot \color{blue}{\left({b}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

   13. Applied rewrites 50.0%

       \[\leadsto \left(\left(b \cdot b\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 41.4% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-313}:\\ \;\;\;\;\left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(t\_0 + t\_0\right) + \sin \left(\mathsf{fma}\left(angle\_m \cdot \pi, 0.005555555555555556, \left(-0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle\_m \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.005555555555555556)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-313)
      (*
       (* -2.0 (exp (* (log a) 2.0)))
       (/
        (+
         (sin (+ t_0 t_0))
         (sin
          (fma
           (* angle_m PI)
           0.005555555555555556
           (* (* -0.005555555555555556 angle_m) PI))))
        2.0))
      (*
       (* (* b b) 2.0)
       (*
        (sin (* (fma 0.011111111111111112 (* angle_m PI) PI) 0.5))
        (sin t_0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-313) {
		tmp = (-2.0 * exp((log(a) * 2.0))) * ((sin((t_0 + t_0)) + sin(fma((angle_m * ((double) M_PI)), 0.005555555555555556, ((-0.005555555555555556 * angle_m) * ((double) M_PI))))) / 2.0);
	} else {
		tmp = ((b * b) * 2.0) * (sin((fma(0.011111111111111112, (angle_m * ((double) M_PI)), ((double) M_PI)) * 0.5)) * sin(t_0));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-313)
		tmp = Float64(Float64(-2.0 * exp(Float64(log(a) * 2.0))) * Float64(Float64(sin(Float64(t_0 + t_0)) + sin(fma(Float64(angle_m * pi), 0.005555555555555556, Float64(Float64(-0.005555555555555556 * angle_m) * pi)))) / 2.0));
	else
		tmp = Float64(Float64(Float64(b * b) * 2.0) * Float64(sin(Float64(fma(0.011111111111111112, Float64(angle_m * pi), pi) * 0.5)) * sin(t_0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-313], N[(N[(-2.0 * N[Exp[N[(N[Log[a], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(t$95$0 + t$95$0), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556 + N[(N[(-0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision] + Pi), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000001e-313

   1. Initial program 55.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \]

   5. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 26.6

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

   7. Applied rewrites 26.6%

      \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

         \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      15. sin-cos-mult N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\color{blue}{2}} \]

   9. Applied rewrites 25.5%

      \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       3. lift-fma.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       4. associate-*r* N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       5. *-commutative N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{1}{180}, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \mathsf{PI}\left(\right), \frac{1}{180}, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       8. lift-PI.f64 27.2

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, 0.005555555555555556, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}{2} \]

       9. lift-neg.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

       10. lift-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

       11. lift-PI.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

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

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)\right)\right)}{2} \]

       14. metadata-eval N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)}{2} \]

       15. *-commutative N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2} \]

       16. associate-*r* N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{2} \]

       17. lower-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{2} \]

       18. lower-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, \frac{1}{180}, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{2} \]

       19. lift-PI.f64 27.2

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, 0.005555555555555556, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2} \]

   11. Applied rewrites 27.2%

       \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, 0.005555555555555556, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2} \]

   if 1.00000000001e-313 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 51.2%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 65.4%

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

   11. Taylor expanded in a around 0

       \[\leadsto 2 \cdot \color{blue}{\left({b}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

   13. Applied rewrites 50.0%

       \[\leadsto \left(\left(b \cdot b\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-313}:\\ \;\;\;\;\left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(angle \cdot \pi, 0.005555555555555556, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.1% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-313}:\\ \;\;\;\;\left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(t\_0 + t\_0\right) + \sin \left(\mathsf{fma}\left(angle\_m, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle\_m \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.005555555555555556)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-313)
      (*
       (* -2.0 (exp (* (log a) 2.0)))
       (/
        (+
         (sin (+ t_0 t_0))
         (sin
          (fma
           angle_m
           (* 0.005555555555555556 PI)
           (* (* -0.005555555555555556 angle_m) PI))))
        2.0))
      (*
       (* (* b b) 2.0)
       (*
        (sin (* (fma 0.011111111111111112 (* angle_m PI) PI) 0.5))
        (sin t_0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-313) {
		tmp = (-2.0 * exp((log(a) * 2.0))) * ((sin((t_0 + t_0)) + sin(fma(angle_m, (0.005555555555555556 * ((double) M_PI)), ((-0.005555555555555556 * angle_m) * ((double) M_PI))))) / 2.0);
	} else {
		tmp = ((b * b) * 2.0) * (sin((fma(0.011111111111111112, (angle_m * ((double) M_PI)), ((double) M_PI)) * 0.5)) * sin(t_0));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-313)
		tmp = Float64(Float64(-2.0 * exp(Float64(log(a) * 2.0))) * Float64(Float64(sin(Float64(t_0 + t_0)) + sin(fma(angle_m, Float64(0.005555555555555556 * pi), Float64(Float64(-0.005555555555555556 * angle_m) * pi)))) / 2.0));
	else
		tmp = Float64(Float64(Float64(b * b) * 2.0) * Float64(sin(Float64(fma(0.011111111111111112, Float64(angle_m * pi), pi) * 0.5)) * sin(t_0)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-313], N[(N[(-2.0 * N[Exp[N[(N[Log[a], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(t$95$0 + t$95$0), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision] + N[(N[(-0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision] + Pi), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000001e-313

   1. Initial program 55.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \]

   5. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 26.6

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

   7. Applied rewrites 26.6%

      \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

         \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      15. sin-cos-mult N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{\color{blue}{2}} \]

   9. Applied rewrites 25.5%

      \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}{\color{blue}{2}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       3. lift-fma.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       4. associate-*r* N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       5. *-commutative N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180} + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       6. associate-*l* N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       7. *-commutative N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) + \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \mathsf{PI}\left(\right), -\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)}{2} \]

       10. lift-PI.f64 26.9

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, -\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}{2} \]

       11. lift-neg.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

       13. lift-PI.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

       14. lift-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \mathsf{neg}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right)}{2} \]

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

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)\right)\right)}{2} \]

       16. metadata-eval N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)}{2} \]

       17. *-commutative N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{2} \]

       18. associate-*r* N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{2} \]

       19. lower-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{2} \]

       20. lower-*.f64 N/A

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(-\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) + \sin \left(\mathsf{fma}\left(angle, \frac{1}{180} \cdot \pi, \left(\frac{-1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{2} \]

       21. lift-PI.f64 26.5

           \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2} \]

   11. Applied rewrites 26.5%

       \[\leadsto \left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(-\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \sin \left(\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2} \]

   if 1.00000000001e-313 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 51.2%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 65.4%

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

   11. Taylor expanded in a around 0

       \[\leadsto 2 \cdot \color{blue}{\left({b}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

   13. Applied rewrites 50.0%

       \[\leadsto \left(\left(b \cdot b\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-313}:\\ \;\;\;\;\left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 + \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(angle, 0.005555555555555556 \cdot \pi, \left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.2% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-313}:\\ \;\;\;\;\left(-2 \cdot e^{\log a \cdot 2}\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle\_m \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-313)
      (* (* -2.0 (exp (* (log a) 2.0))) (* (sin t_0) (cos t_0)))
      (*
       (* (* b b) 2.0)
       (*
        (sin (* (fma 0.011111111111111112 (* angle_m PI) PI) 0.5))
        (sin (* (* angle_m PI) 0.005555555555555556))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-313) {
		tmp = (-2.0 * exp((log(a) * 2.0))) * (sin(t_0) * cos(t_0));
	} else {
		tmp = ((b * b) * 2.0) * (sin((fma(0.011111111111111112, (angle_m * ((double) M_PI)), ((double) M_PI)) * 0.5)) * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-313)
		tmp = Float64(Float64(-2.0 * exp(Float64(log(a) * 2.0))) * Float64(sin(t_0) * cos(t_0)));
	else
		tmp = Float64(Float64(Float64(b * b) * 2.0) * Float64(sin(Float64(fma(0.011111111111111112, Float64(angle_m * pi), pi) * 0.5)) * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-313], N[(N[(-2.0 * N[Exp[N[(N[Log[a], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision] + Pi), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000001e-313

   1. Initial program 55.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \]

   5. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 26.6

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

   7. Applied rewrites 26.6%

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

   if 1.00000000001e-313 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 51.2%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. sin-+PI/2-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sum-sin N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 65.4%

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

   11. Taylor expanded in a around 0

       \[\leadsto 2 \cdot \color{blue}{\left({b}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

   13. Applied rewrites 50.0%

       \[\leadsto \left(\left(b \cdot b\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 35.8% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle\_m \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \end{array} \]

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 b) 2.0)
   (*
    (sin (* (fma 0.011111111111111112 (* angle_m PI) PI) 0.5))
    (sin (* (* angle_m PI) 0.005555555555555556))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (((b * b) * 2.0) * (sin((fma(0.011111111111111112, (angle_m * ((double) M_PI)), ((double) M_PI)) * 0.5)) * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556))));
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(b * b) * 2.0) * Float64(sin(Float64(fma(0.011111111111111112, Float64(angle_m * pi), pi) * 0.5)) * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)))))
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision] + Pi), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in angle around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. difference-of-squares N/A

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

   14. lower-*.f64 N/A

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

   15. lower-+.f64 N/A

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

   16. lower--.f64 55.2

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

5. Applied rewrites 55.2%

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

6. Taylor expanded in angle around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 65.4%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. count-2-rev N/A

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

   8. sin-+PI/2-rev N/A

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

   9. sin-+PI/2-rev N/A

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

   10. sum-sin N/A

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

   11. lower-*.f64 N/A

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

10. Applied rewrites 66.1%

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

11. Taylor expanded in a around 0

    \[\leadsto 2 \cdot \color{blue}{\left({b}^{2} \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) + \frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

13. Applied rewrites 30.2%

    \[\leadsto \left(\left(b \cdot b\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.011111111111111112, angle \cdot \pi, \pi\right) \cdot 0.5\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))