ab-angle->ABCF B

Percentage Accurate: 54.1% → 65.7%

Time: 8.3s

Alternatives: 22

Speedup: 2.2×

• 32.6% 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: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_1 := \left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\\ t_2 := \frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, angle\_m, t\_1\right)\right)}{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 6 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, t\_2, \left(t\_2 \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sin t\_0 \cdot b\right) \cdot b\right) \cdot 2\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) + \frac{\pi}{2}\right)\\ \end{array} \end{array} \end{array} \]

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 (* (* angle_m PI) -0.005555555555555556))
        (t_2
         (/
          (+
           (sin (- t_0 t_1))
           (sin (fma (* 0.005555555555555556 PI) angle_m t_1)))
          2.0)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 6e+140)
      (fma
       (fma (* -2.0 a) t_2 (* (* t_2 (* 0.0 b)) 2.0))
       a
       (* (* (* b b) 2.0) t_2))
      (*
       (* (* (* (sin t_0) b) b) 2.0)
       (sin (+ (- (* PI (* 0.005555555555555556 angle_m))) (/ PI 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_1 = (angle_m * ((double) M_PI)) * -0.005555555555555556;
	double t_2 = (sin((t_0 - t_1)) + sin(fma((0.005555555555555556 * ((double) M_PI)), angle_m, t_1))) / 2.0;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 6e+140) {
		tmp = fma(fma((-2.0 * a), t_2, ((t_2 * (0.0 * b)) * 2.0)), a, (((b * b) * 2.0) * t_2));
	} else {
		tmp = (((sin(t_0) * b) * b) * 2.0) * sin((-(((double) M_PI) * (0.005555555555555556 * angle_m)) + (((double) M_PI) / 2.0)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_1 = Float64(Float64(angle_m * pi) * -0.005555555555555556)
	t_2 = Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(0.005555555555555556 * pi), angle_m, t_1))) / 2.0)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 6e+140)
		tmp = fma(fma(Float64(-2.0 * a), t_2, Float64(Float64(t_2 * Float64(0.0 * b)) * 2.0)), a, Float64(Float64(Float64(b * b) * 2.0) * t_2));
	else
		tmp = Float64(Float64(Float64(Float64(sin(t_0) * b) * b) * 2.0) * sin(Float64(Float64(-Float64(pi * Float64(0.005555555555555556 * angle_m))) + Float64(pi / 2.0))));
	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]}, Block[{t$95$1 = N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle$95$m + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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], 6e+140], N[(N[(N[(-2.0 * a), $MachinePrecision] * t$95$2 + N[(N[(t$95$2 * N[(0.0 * b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[((-N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]) + N[(Pi / 2.0), $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)))) < 5.99999999999999993e140

   1. Initial program 59.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 59.0

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

   3. Applied rewrites 59.0%

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

   4. Taylor expanded in angle around inf

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

   6. Applied rewrites 59.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), \color{blue}{a}, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       12. sin-cos-mult N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       13. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       12. sin-cos-mult N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       13. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

   13. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)\right)\right) \]

       12. sin-cos-mult N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)}{2}\right) \]

       13. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180} - \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) - \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{180}\right)}{2}\right) \]

   15. Applied rewrites 65.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)}{2}, \left(\frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)}{2} \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \frac{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556 - \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot \pi, angle, \left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)\right)}{2}\right) \]

   if 5.99999999999999993e140 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 43.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 43.3

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

   3. Applied rewrites 43.3%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.7

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

   6. Applied rewrites 50.7%

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

   8. Applied rewrites 65.2%

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

Alternative 2: 65.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) + \frac{\pi}{2}\right)\\ t_1 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_2 := \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+298}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(a \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right), -2, \left(\pi \cdot a\right) \cdot -0.011111111111111112\right) \cdot angle\_m, a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot t\_2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+234}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - \frac{1}{\frac{1}{a \cdot a}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_2 \cdot b\right) \cdot b\right) \cdot 2\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (+ (- (* PI (* 0.005555555555555556 angle_m))) (/ PI 2.0))))
        (t_1 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_2 (sin (* (* angle_m PI) 0.005555555555555556))))
   (*
    angle_s
    (if (<= t_1 -2e+298)
      (fma
       (*
        (fma
         (*
          (* a (* angle_m angle_m))
          (* (* (* PI PI) PI) -1.1431184270690443e-7))
         -2.0
         (* (* PI a) -0.011111111111111112))
        angle_m)
       a
       (*
        (* (* b b) 2.0)
        (* (cos (* (* angle_m PI) -0.005555555555555556)) t_2)))
      (if (<= t_1 1e+234)
        (*
         (*
          (* 2.0 (- (pow b 2.0) (/ 1.0 (/ 1.0 (* a a)))))
          (sin (* PI (/ angle_m 180.0))))
         t_0)
        (* (* (* (* t_2 b) b) 2.0) t_0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((-(((double) M_PI) * (0.005555555555555556 * angle_m)) + (((double) M_PI) / 2.0)));
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_2 = sin(((angle_m * ((double) M_PI)) * 0.005555555555555556));
	double tmp;
	if (t_1 <= -2e+298) {
		tmp = fma((fma(((a * (angle_m * angle_m)) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7)), -2.0, ((((double) M_PI) * a) * -0.011111111111111112)) * angle_m), a, (((b * b) * 2.0) * (cos(((angle_m * ((double) M_PI)) * -0.005555555555555556)) * t_2)));
	} else if (t_1 <= 1e+234) {
		tmp = ((2.0 * (pow(b, 2.0) - (1.0 / (1.0 / (a * a))))) * sin((((double) M_PI) * (angle_m / 180.0)))) * t_0;
	} else {
		tmp = (((t_2 * b) * b) * 2.0) * t_0;
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(Float64(-Float64(pi * Float64(0.005555555555555556 * angle_m))) + Float64(pi / 2.0)))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_2 = sin(Float64(Float64(angle_m * pi) * 0.005555555555555556))
	tmp = 0.0
	if (t_1 <= -2e+298)
		tmp = fma(Float64(fma(Float64(Float64(a * Float64(angle_m * angle_m)) * Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7)), -2.0, Float64(Float64(pi * a) * -0.011111111111111112)) * angle_m), a, Float64(Float64(Float64(b * b) * 2.0) * Float64(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)) * t_2)));
	elseif (t_1 <= 1e+234)
		tmp = Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - Float64(1.0 / Float64(1.0 / Float64(a * a))))) * sin(Float64(pi * Float64(angle_m / 180.0)))) * t_0);
	else
		tmp = Float64(Float64(Float64(Float64(t_2 * b) * b) * 2.0) * 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[Sin[N[((-N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, -2e+298], N[(N[(N[(N[(N[(a * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(Pi * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * a + N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+234], N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[(1.0 / N[(1.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(t$95$2 * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -1.9999999999999999e298

   1. Initial program 51.6%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 50.8

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

   3. Applied rewrites 50.8%

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

   4. Taylor expanded in angle around inf

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

   6. Applied rewrites 50.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 73.8%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{fma}\left(angle \cdot \left(-2 \cdot \left(a \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(-2 \cdot \left(a \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle, a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(-2 \cdot \left(a \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) + \frac{-1}{90} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle, a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

   12. Applied rewrites 72.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(a \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right), -2, \left(\pi \cdot a\right) \cdot -0.011111111111111112\right) \cdot angle, a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]

   if -1.9999999999999999e298 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000000000002e234

   1. Initial program 61.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 61.3

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

   3. Applied rewrites 61.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if 1.00000000000000002e234 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 42.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 41.8

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

   3. Applied rewrites 41.8%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.5

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

   6. Applied rewrites 50.5%

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

   8. Applied rewrites 67.4%

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

Alternative 3: 65.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, t\_0, \left(t\_0 \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right) \cdot b\right) \cdot 2\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) + \frac{\pi}{2}\right)\\ \end{array} \end{array} \end{array} \]

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
         (*
          (cos (* (* angle_m PI) -0.005555555555555556))
          (sin (* (* 0.005555555555555556 angle_m) PI)))))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 2e+52)
      (fma
       (fma (* -2.0 a) t_0 (* (* t_0 (* 0.0 b)) 2.0))
       a
       (* (* (* b b) 2.0) t_0))
      (*
       (* (* (* (sin (* (* angle_m PI) 0.005555555555555556)) b) b) 2.0)
       (sin (+ (- (* PI (* 0.005555555555555556 angle_m))) (/ PI 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = cos(((angle_m * ((double) M_PI)) * -0.005555555555555556)) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)));
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 2e+52) {
		tmp = fma(fma((-2.0 * a), t_0, ((t_0 * (0.0 * b)) * 2.0)), a, (((b * b) * 2.0) * t_0));
	} else {
		tmp = (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b) * b) * 2.0) * sin((-(((double) M_PI) * (0.005555555555555556 * angle_m)) + (((double) M_PI) / 2.0)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)))
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 2e+52)
		tmp = fma(fma(Float64(-2.0 * a), t_0, Float64(Float64(t_0 * Float64(0.0 * b)) * 2.0)), a, Float64(Float64(Float64(b * b) * 2.0) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * sin(Float64(Float64(-Float64(pi * Float64(0.005555555555555556 * angle_m))) + Float64(pi / 2.0))));
	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[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $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], 2e+52], N[(N[(N[(-2.0 * a), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * N[(0.0 * b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[((-N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]) + N[(Pi / 2.0), $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)))) < 2e52

   1. Initial program 59.6%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 59.1

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

   3. Applied rewrites 59.1%

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

   4. Taylor expanded in angle around inf

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

   6. Applied rewrites 59.3%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 66.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), \color{blue}{a}, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       8. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       12. mult-flip N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       16. lift-PI.f64 66.4

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]

   11. Applied rewrites 66.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       8. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       12. mult-flip N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       16. lift-PI.f64 66.4

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]

   13. Applied rewrites 66.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       8. mult-flip N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       11. lift-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       12. mult-flip N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{180}\right) \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       16. lift-PI.f64 66.4

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \]

   15. Applied rewrites 66.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right) \]

   if 2e52 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 45.3%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 44.7

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

   3. Applied rewrites 44.7%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 51.3

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

   6. Applied rewrites 51.3%

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

   8. Applied rewrites 64.3%

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

Alternative 4: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) + \frac{\pi}{2}\right)\\ t_1 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+304}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+234}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - \frac{1}{\frac{1}{a \cdot a}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right) \cdot b\right) \cdot 2\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (+ (- (* PI (* 0.005555555555555556 angle_m))) (/ PI 2.0))))
        (t_1 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_1 -4e+304)
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_1 1e+234)
        (*
         (*
          (* 2.0 (- (pow b 2.0) (/ 1.0 (/ 1.0 (* a a)))))
          (sin (* PI (/ angle_m 180.0))))
         t_0)
        (*
         (* (* (* (sin (* (* angle_m PI) 0.005555555555555556)) b) b) 2.0)
         t_0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((-(((double) M_PI) * (0.005555555555555556 * angle_m)) + (((double) M_PI) / 2.0)));
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_1 <= -4e+304) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_1 <= 1e+234) {
		tmp = ((2.0 * (pow(b, 2.0) - (1.0 / (1.0 / (a * a))))) * sin((((double) M_PI) * (angle_m / 180.0)))) * t_0;
	} else {
		tmp = (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b) * b) * 2.0) * t_0;
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.sin((-(Math.PI * (0.005555555555555556 * angle_m)) + (Math.PI / 2.0)));
	double t_1 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if (t_1 <= -4e+304) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else if (t_1 <= 1e+234) {
		tmp = ((2.0 * (Math.pow(b, 2.0) - (1.0 / (1.0 / (a * a))))) * Math.sin((Math.PI * (angle_m / 180.0)))) * t_0;
	} else {
		tmp = (((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * b) * b) * 2.0) * t_0;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.sin((-(math.pi * (0.005555555555555556 * angle_m)) + (math.pi / 2.0)))
	t_1 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	tmp = 0
	if t_1 <= -4e+304:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	elif t_1 <= 1e+234:
		tmp = ((2.0 * (math.pow(b, 2.0) - (1.0 / (1.0 / (a * a))))) * math.sin((math.pi * (angle_m / 180.0)))) * t_0
	else:
		tmp = (((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * b) * b) * 2.0) * t_0
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(Float64(-Float64(pi * Float64(0.005555555555555556 * angle_m))) + Float64(pi / 2.0)))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -4e+304)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_1 <= 1e+234)
		tmp = Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - Float64(1.0 / Float64(1.0 / Float64(a * a))))) * sin(Float64(pi * Float64(angle_m / 180.0)))) * t_0);
	else
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * t_0);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = sin((-(pi * (0.005555555555555556 * angle_m)) + (pi / 2.0)));
	t_1 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	tmp = 0.0;
	if (t_1 <= -4e+304)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	elseif (t_1 <= 1e+234)
		tmp = ((2.0 * ((b ^ 2.0) - (1.0 / (1.0 / (a * a))))) * sin((pi * (angle_m / 180.0)))) * t_0;
	else
		tmp = (((sin(((angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[((-N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, -4e+304], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+234], N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[(1.0 / N[(1.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -3.9999999999999998e304

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.0

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.0%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.1

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.1%

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

   if -3.9999999999999998e304 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000000000002e234

   1. Initial program 61.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 61.3

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

   3. Applied rewrites 61.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if 1.00000000000000002e234 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 42.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 41.8

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

   3. Applied rewrites 41.8%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.5

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

   6. Applied rewrites 50.5%

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

   8. Applied rewrites 67.4%

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

Alternative 5: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - \frac{1}{\frac{1}{a \cdot a}}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right) \cdot b\right) \cdot 2\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) + \frac{\pi}{2}\right)\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_1 (- INFINITY))
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_1 2e+219)
        (*
         (* (* 2.0 (- (pow b 2.0) (/ 1.0 (/ 1.0 (* a a))))) (sin t_0))
         (cos t_0))
        (*
         (* (* (* (sin (* (* angle_m PI) 0.005555555555555556)) b) b) 2.0)
         (sin (+ (- (* PI (* 0.005555555555555556 angle_m))) (/ PI 2.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_1 <= 2e+219) {
		tmp = ((2.0 * (pow(b, 2.0) - (1.0 / (1.0 / (a * a))))) * sin(t_0)) * cos(t_0);
	} else {
		tmp = (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b) * b) * 2.0) * sin((-(((double) M_PI) * (0.005555555555555556 * angle_m)) + (((double) M_PI) / 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double t_1 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else if (t_1 <= 2e+219) {
		tmp = ((2.0 * (Math.pow(b, 2.0) - (1.0 / (1.0 / (a * a))))) * Math.sin(t_0)) * Math.cos(t_0);
	} else {
		tmp = (((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * b) * b) * 2.0) * Math.sin((-(Math.PI * (0.005555555555555556 * angle_m)) + (Math.PI / 2.0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 180.0)
	t_1 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	elif t_1 <= 2e+219:
		tmp = ((2.0 * (math.pow(b, 2.0) - (1.0 / (1.0 / (a * a))))) * math.sin(t_0)) * math.cos(t_0)
	else:
		tmp = (((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * b) * b) * 2.0) * math.sin((-(math.pi * (0.005555555555555556 * angle_m)) + (math.pi / 2.0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_1 <= 2e+219)
		tmp = Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - Float64(1.0 / Float64(1.0 / Float64(a * a))))) * sin(t_0)) * cos(t_0));
	else
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * sin(Float64(Float64(-Float64(pi * Float64(0.005555555555555556 * angle_m))) + Float64(pi / 2.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	t_1 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	elseif (t_1 <= 2e+219)
		tmp = ((2.0 * ((b ^ 2.0) - (1.0 / (1.0 / (a * a))))) * sin(t_0)) * cos(t_0);
	else
		tmp = (((sin(((angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * sin((-(pi * (0.005555555555555556 * angle_m)) + (pi / 2.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+219], N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[(1.0 / N[(1.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[((-N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.3%

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

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.99999999999999993e219

   1. Initial program 61.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.8

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

   3. Applied rewrites 61.8%

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

   if 1.99999999999999993e219 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 42.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 42.1

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.6

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

   6. Applied rewrites 50.6%

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

   8. Applied rewrites 67.0%

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

Alternative 6: 65.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(\left(2 \cdot \left(\frac{1}{\frac{1}{b \cdot b}} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right) \cdot b\right) \cdot 2\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) + \frac{\pi}{2}\right)\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_1 (- INFINITY))
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_1 2e+219)
        (*
         (* (* 2.0 (- (/ 1.0 (/ 1.0 (* b b))) (pow a 2.0))) (sin t_0))
         (cos t_0))
        (*
         (* (* (* (sin (* (* angle_m PI) 0.005555555555555556)) b) b) 2.0)
         (sin (+ (- (* PI (* 0.005555555555555556 angle_m))) (/ PI 2.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_1 <= 2e+219) {
		tmp = ((2.0 * ((1.0 / (1.0 / (b * b))) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
	} else {
		tmp = (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b) * b) * 2.0) * sin((-(((double) M_PI) * (0.005555555555555556 * angle_m)) + (((double) M_PI) / 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double t_1 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else if (t_1 <= 2e+219) {
		tmp = ((2.0 * ((1.0 / (1.0 / (b * b))) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
	} else {
		tmp = (((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * b) * b) * 2.0) * Math.sin((-(Math.PI * (0.005555555555555556 * angle_m)) + (Math.PI / 2.0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 180.0)
	t_1 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	elif t_1 <= 2e+219:
		tmp = ((2.0 * ((1.0 / (1.0 / (b * b))) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)
	else:
		tmp = (((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * b) * b) * 2.0) * math.sin((-(math.pi * (0.005555555555555556 * angle_m)) + (math.pi / 2.0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_1 <= 2e+219)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(1.0 / Float64(1.0 / Float64(b * b))) - (a ^ 2.0))) * sin(t_0)) * cos(t_0));
	else
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * sin(Float64(Float64(-Float64(pi * Float64(0.005555555555555556 * angle_m))) + Float64(pi / 2.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	t_1 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	elseif (t_1 <= 2e+219)
		tmp = ((2.0 * ((1.0 / (1.0 / (b * b))) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
	else
		tmp = (((sin(((angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * sin((-(pi * (0.005555555555555556 * angle_m)) + (pi / 2.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+219], N[(N[(N[(2.0 * N[(N[(1.0 / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[((-N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.3%

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

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.99999999999999993e219

   1. Initial program 61.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-neg N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.9

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

   3. Applied rewrites 61.9%

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

   if 1.99999999999999993e219 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 42.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 42.1

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

   3. Applied rewrites 42.1%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.6

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

   6. Applied rewrites 50.6%

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

   8. Applied rewrites 67.0%

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

Alternative 7: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(\sin t\_1 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot 2\right) \cdot \cos t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right) \cdot b\right) \cdot 2\right) \cdot \sin \left(\left(-t\_1\right) + \frac{\pi}{2}\right)\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_1 (* PI (* 0.005555555555555556 angle_m))))
   (*
    angle_s
    (if (<= t_0 (- INFINITY))
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 2e+146)
        (* (* (* (sin t_1) (- (* b b) (* a a))) 2.0) (cos t_1))
        (*
         (* (* (* (sin (* (* angle_m PI) 0.005555555555555556)) b) b) 2.0)
         (sin (+ (- t_1) (/ PI 2.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 2e+146) {
		tmp = ((sin(t_1) * ((b * b) - (a * a))) * 2.0) * cos(t_1);
	} else {
		tmp = (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b) * b) * 2.0) * sin((-t_1 + (((double) M_PI) / 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double t_1 = Math.PI * (0.005555555555555556 * angle_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else if (t_0 <= 2e+146) {
		tmp = ((Math.sin(t_1) * ((b * b) - (a * a))) * 2.0) * Math.cos(t_1);
	} else {
		tmp = (((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * b) * b) * 2.0) * Math.sin((-t_1 + (Math.PI / 2.0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	t_1 = math.pi * (0.005555555555555556 * angle_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	elif t_0 <= 2e+146:
		tmp = ((math.sin(t_1) * ((b * b) - (a * a))) * 2.0) * math.cos(t_1)
	else:
		tmp = (((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * b) * b) * 2.0) * math.sin((-t_1 + (math.pi / 2.0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 2e+146)
		tmp = Float64(Float64(Float64(sin(t_1) * Float64(Float64(b * b) - Float64(a * a))) * 2.0) * cos(t_1));
	else
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * sin(Float64(Float64(-t_1) + Float64(pi / 2.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	t_1 = pi * (0.005555555555555556 * angle_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	elseif (t_0 <= 2e+146)
		tmp = ((sin(t_1) * ((b * b) - (a * a))) * 2.0) * cos(t_1);
	else
		tmp = (((sin(((angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0) * sin((-t_1 + (pi / 2.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+146], N[(N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[((-t$95$1) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.3%

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

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.99999999999999987e146

   1. Initial program 62.6%

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

   3. Applied rewrites 62.5%

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

   if 1.99999999999999987e146 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 43.8%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 43.2

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

   3. Applied rewrites 43.2%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.7

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

   6. Applied rewrites 50.7%

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

   8. Applied rewrites 65.2%

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

Alternative 8: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\left(\left(\sin t\_1 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot 2\right) \cdot \cos t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right) \cdot b\right) \cdot 2\right)\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
        (t_1 (* PI (* 0.005555555555555556 angle_m))))
   (*
    angle_s
    (if (<= t_0 (- INFINITY))
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 2e+127)
        (* (* (* (sin t_1) (- (* b b) (* a a))) 2.0) (cos t_1))
        (*
         (cos (* (* angle_m PI) -0.005555555555555556))
         (*
          (* (* (sin (* (* angle_m PI) 0.005555555555555556)) b) b)
          2.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 2e+127) {
		tmp = ((sin(t_1) * ((b * b) - (a * a))) * 2.0) * cos(t_1);
	} else {
		tmp = cos(((angle_m * ((double) M_PI)) * -0.005555555555555556)) * (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b) * b) * 2.0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double t_1 = Math.PI * (0.005555555555555556 * angle_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else if (t_0 <= 2e+127) {
		tmp = ((Math.sin(t_1) * ((b * b) - (a * a))) * 2.0) * Math.cos(t_1);
	} else {
		tmp = Math.cos(((angle_m * Math.PI) * -0.005555555555555556)) * (((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * b) * b) * 2.0);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	t_1 = math.pi * (0.005555555555555556 * angle_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	elif t_0 <= 2e+127:
		tmp = ((math.sin(t_1) * ((b * b) - (a * a))) * 2.0) * math.cos(t_1)
	else:
		tmp = math.cos(((angle_m * math.pi) * -0.005555555555555556)) * (((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * b) * b) * 2.0)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 2e+127)
		tmp = Float64(Float64(Float64(sin(t_1) * Float64(Float64(b * b) - Float64(a * a))) * 2.0) * cos(t_1));
	else
		tmp = Float64(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)) * Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	t_1 = pi * (0.005555555555555556 * angle_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	elseif (t_0 <= 2e+127)
		tmp = ((sin(t_1) * ((b * b) - (a * a))) * 2.0) * cos(t_1);
	else
		tmp = cos(((angle_m * pi) * -0.005555555555555556)) * (((sin(((angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+127], N[(N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.3%

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

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.99999999999999991e127

   1. Initial program 62.7%

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

   3. Applied rewrites 62.6%

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

   if 1.99999999999999991e127 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 44.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 43.5

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

   3. Applied rewrites 43.5%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.8

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

   6. Applied rewrites 50.8%

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

   8. Applied rewrites 64.9%

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

Alternative 9: 64.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{-258}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot b\right) \cdot b\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -5e-258)
    (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
    (*
     (cos (* (* angle_m PI) -0.005555555555555556))
     (* (* (* (sin (* (* angle_m PI) 0.005555555555555556)) b) b) 2.0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -5e-258) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else {
		tmp = cos(((angle_m * ((double) M_PI)) * -0.005555555555555556)) * (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * b) * b) * 2.0);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -5e-258) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else {
		tmp = Math.cos(((angle_m * Math.PI) * -0.005555555555555556)) * (((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * b) * b) * 2.0);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -5e-258:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	else:
		tmp = math.cos(((angle_m * math.pi) * -0.005555555555555556)) * (((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * b) * b) * 2.0)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -5e-258)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	else
		tmp = Float64(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)) * Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -5e-258)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	else
		tmp = cos(((angle_m * pi) * -0.005555555555555556)) * (((sin(((angle_m * pi) * 0.005555555555555556)) * b) * b) * 2.0);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-258], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $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)))) < -4.9999999999999999e-258

   1. Initial program 54.3%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.8

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 50.9

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 50.9%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 61.2

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 61.2%

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

   if -4.9999999999999999e-258 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 53.4

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

   3. Applied rewrites 53.4%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 57.5

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

   6. Applied rewrites 57.5%

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

   8. Applied rewrites 66.0%

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

Alternative 10: 62.3% accurate, 1.3× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) (- INFINITY))
    (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
    (*
     (sin (* 2.0 (* (* 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) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else {
		tmp = sin((2.0 * ((angle_m * ((double) M_PI)) * 0.005555555555555556))) * ((a + b) * (b - a));
	}
	return angle_s * tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.3%

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

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

   1. Initial program 54.7%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 54.2

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

   3. Applied rewrites 54.2%

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

   4. Taylor expanded in angle around inf

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

   6. Applied rewrites 59.6%

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

   8. Applied rewrites 59.6%

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

Alternative 11: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 (- INFINITY))
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 2e+300)
        (*
         (* (sin (* (* PI angle_m) 0.005555555555555556)) 2.0)
         (* (+ a b) (- b a)))
        (*
         (* (* (* (* PI (+ a b)) (- b a)) angle_m) 0.011111111111111112)
         (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* PI PI) 1.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 2e+300) {
		tmp = (sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * 2.0) * ((a + b) * (b - a));
	} else {
		tmp = ((((((double) M_PI) * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112) * fma((-1.54320987654321e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 1.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 2e+300)
		tmp = Float64(Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * 2.0) * Float64(Float64(a + b) * Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(pi * Float64(a + b)) * Float64(b - a)) * angle_m) * 0.011111111111111112) * fma(Float64(-1.54320987654321e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 1.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.3%

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

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.0000000000000001e300

   1. Initial program 61.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 60.9

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in angle around inf

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

   6. Applied rewrites 61.3%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift--.f64 59.4

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

   9. Applied rewrites 59.4%

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

   if 2.0000000000000001e300 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 40.5%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 37.9

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 37.9

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\color{blue}{\mathsf{PI}\left(\right)}}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lift-PI.f64 39.6

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   7. Applied rewrites 39.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      2. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      4. difference-of-squares-rev N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      14. lift--.f64 54.3

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 54.3%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+304}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\left(\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 -4e+304)
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 5e-264)
        (* (* (* -2.0 (* a a)) (sin (* PI (/ angle_m 180.0)))) 1.0)
        (*
         (* (* (* (* PI (+ a b)) (- b a)) angle_m) 0.011111111111111112)
         (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* PI PI) 1.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -4e+304) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 5e-264) {
		tmp = ((-2.0 * (a * a)) * sin((((double) M_PI) * (angle_m / 180.0)))) * 1.0;
	} else {
		tmp = ((((((double) M_PI) * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112) * fma((-1.54320987654321e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 1.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -4e+304)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 5e-264)
		tmp = Float64(Float64(Float64(-2.0 * Float64(a * a)) * sin(Float64(pi * Float64(angle_m / 180.0)))) * 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(pi * Float64(a + b)) * Float64(b - a)) * angle_m) * 0.011111111111111112) * fma(Float64(-1.54320987654321e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 1.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -4e+304], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-264], N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -3.9999999999999998e304

   1. Initial program 51.4%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.0

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.0%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 74.1

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.1%

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

   if -3.9999999999999998e304 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 5.0000000000000001e-264

   1. Initial program 65.0%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 63.1%

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

   5. Taylor expanded in a around inf

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

      1. pow-to-exp N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 62.5

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

   7. Applied rewrites 62.5%

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

   if 5.0000000000000001e-264 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 48.1%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 44.2

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 44.3

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied rewrites 44.3%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\color{blue}{\mathsf{PI}\left(\right)}}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lift-PI.f64 44.9

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   7. Applied rewrites 44.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      2. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      4. difference-of-squares-rev N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      14. lift--.f64 52.7

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 52.7%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-258}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 -5e-258)
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 2e+300)
        (*
         (* (* (sin (* (* angle_m PI) 0.005555555555555556)) (* b b)) 2.0)
         1.0)
        (*
         (* (* (* (* PI (+ a b)) (- b a)) angle_m) 0.011111111111111112)
         (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* PI PI) 1.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -5e-258) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 2e+300) {
		tmp = ((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * (b * b)) * 2.0) * 1.0;
	} else {
		tmp = ((((((double) M_PI) * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112) * fma((-1.54320987654321e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 1.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -5e-258)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 2e+300)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * Float64(b * b)) * 2.0) * 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(pi * Float64(a + b)) * Float64(b - a)) * angle_m) * 0.011111111111111112) * fma(Float64(-1.54320987654321e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 1.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -5e-258], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.9999999999999999e-258

   1. Initial program 54.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 51.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 50.8

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.8%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 50.9

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 50.9%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 61.2

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 61.2%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4.9999999999999999e-258 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.0000000000000001e300

   1. Initial program 64.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.1%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \cdot 1 \]
   6. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      5. associate-*l* N/A

         \[\leadsto \left(2 \cdot \left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      6. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{2}\right) \cdot 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{2}\right) \cdot 1 \]

   7. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right)} \cdot 1 \]

   if 2.0000000000000001e300 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 40.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 37.9

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 37.9

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied rewrites 37.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\color{blue}{\mathsf{PI}\left(\right)}}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lift-PI.f64 39.6

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   7. Applied rewrites 39.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      2. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      4. difference-of-squares-rev N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      14. lift--.f64 54.3

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 54.3%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a \cdot a\right)\right) \cdot -2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 (- INFINITY))
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 5e-264)
        (*
         (* (* (sin (* (* angle_m PI) 0.005555555555555556)) (* a a)) -2.0)
         1.0)
        (*
         (* (* (* (* PI (+ a b)) (- b a)) angle_m) 0.011111111111111112)
         (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* PI PI) 1.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 5e-264) {
		tmp = ((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * (a * a)) * -2.0) * 1.0;
	} else {
		tmp = ((((((double) M_PI) * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112) * fma((-1.54320987654321e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 1.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 5e-264)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * Float64(a * a)) * -2.0) * 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(pi * Float64(a + b)) * Float64(b - a)) * angle_m) * 0.011111111111111112) * fma(Float64(-1.54320987654321e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 1.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-264], N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -inf.0

   1. Initial program 51.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.7

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 50.7

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.7%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 74.3

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 74.3%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 5.0000000000000001e-264

   1. Initial program 64.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.0%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1} \]

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \cdot 1 \]
   6. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      5. associate-*l* N/A

         \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot 1 \]

      6. *-commutative N/A

         \[\leadsto \left(\left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{-2}\right) \cdot 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{-2}\right) \cdot 1 \]

   7. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\left(\left(\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a \cdot a\right)\right) \cdot -2\right)} \cdot 1 \]

   if 5.0000000000000001e-264 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 48.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 44.2

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 44.3

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied rewrites 44.3%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\color{blue}{\mathsf{PI}\left(\right)}}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lift-PI.f64 44.9

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   7. Applied rewrites 44.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      2. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      4. difference-of-squares-rev N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      14. lift--.f64 52.7

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 52.7%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 59.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ t_1 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+164}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(0 \cdot t\_0\right), -6.17283950617284 \cdot 10^{-5}, \left(0.011111111111111112 \cdot \pi\right) \cdot \left(1 \cdot t\_0\right)\right) \cdot angle\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a))) (t_1 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_1 -4e+164)
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_1 2e+300)
        (*
         (fma
          (* (* angle_m (* PI PI)) (* 0.0 t_0))
          -6.17283950617284e-5
          (* (* 0.011111111111111112 PI) (* 1.0 t_0)))
         angle_m)
        (*
         (* (* (* (* PI (+ a b)) (- b a)) angle_m) 0.011111111111111112)
         (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* PI PI) 1.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_1 <= -4e+164) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_1 <= 2e+300) {
		tmp = fma(((angle_m * (((double) M_PI) * ((double) M_PI))) * (0.0 * t_0)), -6.17283950617284e-5, ((0.011111111111111112 * ((double) M_PI)) * (1.0 * t_0))) * angle_m;
	} else {
		tmp = ((((((double) M_PI) * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112) * fma((-1.54320987654321e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 1.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(b - a))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -4e+164)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_1 <= 2e+300)
		tmp = Float64(fma(Float64(Float64(angle_m * Float64(pi * pi)) * Float64(0.0 * t_0)), -6.17283950617284e-5, Float64(Float64(0.011111111111111112 * pi) * Float64(1.0 * t_0))) * angle_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(pi * Float64(a + b)) * Float64(b - a)) * angle_m) * 0.011111111111111112) * fma(Float64(-1.54320987654321e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 1.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, -4e+164], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+300], N[(N[(N[(N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(0.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * -6.17283950617284e-5 + N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * N[(1.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision], N[(N[(N[(N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4e164

   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. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.3

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 50.2

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.2%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 50.5

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 50.5%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 66.5

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 66.5%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4e164 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.0000000000000001e300

   1. Initial program 62.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\frac{angle}{180}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

      13. lift-PI.f64 62.1

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \]

   3. Applied rewrites 62.1%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\pi}{2}\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{16200} \cdot \left(angle \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \]

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(angle \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(0 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right), -6.17283950617284 \cdot 10^{-5}, \left(0.011111111111111112 \cdot \pi\right) \cdot \left(1 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot angle} \]

   if 2.0000000000000001e300 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 40.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 37.9

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 37.9

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied rewrites 37.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\color{blue}{\mathsf{PI}\left(\right)}}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lift-PI.f64 39.6

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   7. Applied rewrites 39.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      2. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      4. difference-of-squares-rev N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      14. lift--.f64 54.3

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 54.3%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \left(a \cdot a\right), -0.011111111111111112, \left(\pi \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right) \cdot angle\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 -2e+109)
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 2e+300)
        (*
         (fma
          (* PI (* a a))
          -0.011111111111111112
          (* (* PI (* b b)) 0.011111111111111112))
         angle_m)
        (*
         (* (* (* (* PI (+ a b)) (- b a)) angle_m) 0.011111111111111112)
         (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* PI PI) 1.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -2e+109) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 2e+300) {
		tmp = fma((((double) M_PI) * (a * a)), -0.011111111111111112, ((((double) M_PI) * (b * b)) * 0.011111111111111112)) * angle_m;
	} else {
		tmp = ((((((double) M_PI) * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112) * fma((-1.54320987654321e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 1.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -2e+109)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 2e+300)
		tmp = Float64(fma(Float64(pi * Float64(a * a)), -0.011111111111111112, Float64(Float64(pi * Float64(b * b)) * 0.011111111111111112)) * angle_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(pi * Float64(a + b)) * Float64(b - a)) * angle_m) * 0.011111111111111112) * fma(Float64(-1.54320987654321e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 1.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -2e+109], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(N[(N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112 + N[(N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision], N[(N[(N[(N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -1.99999999999999996e109

   1. Initial program 52.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.4

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 50.3

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 50.5

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 50.5%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 65.2

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 65.2%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -1.99999999999999996e109 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.0000000000000001e300

   1. Initial program 62.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\frac{angle}{180}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

      13. lift-PI.f64 62.5

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \]

   3. Applied rewrites 62.5%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\pi}{2}\right)} \]

   4. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto 2 \cdot \left({b}^{2} \cdot \left(\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \color{blue}{a \cdot \left(-2 \cdot \left(a \cdot \left(\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]

   9. Applied rewrites 62.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), \color{blue}{a}, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto angle \cdot \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{1}{90} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

   12. Applied rewrites 57.8%

       \[\leadsto \mathsf{fma}\left(\pi \cdot \left(a \cdot a\right), -0.011111111111111112, \left(\pi \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right) \cdot angle \]

   if 2.0000000000000001e300 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 40.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 37.9

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 37.9

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied rewrites 37.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\color{blue}{\mathsf{PI}\left(\right)}}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lift-PI.f64 39.6

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   7. Applied rewrites 39.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      2. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      4. difference-of-squares-rev N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{90}}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      14. lift--.f64 54.3

          \[\leadsto \left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 54.3%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)} \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 58.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \left(a \cdot a\right), -0.011111111111111112, \left(\pi \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right) \cdot angle\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 -2e+109)
      (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
      (if (<= t_0 2e+300)
        (*
         (fma
          (* PI (* a a))
          -0.011111111111111112
          (* (* PI (* b b)) 0.011111111111111112))
         angle_m)
        (*
         (* (* (* b b) 2.0) (* PI (* 0.005555555555555556 angle_m)))
         (fma (* -1.54320987654321e-5 (* angle_m angle_m)) (* PI PI) 1.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -2e+109) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else if (t_0 <= 2e+300) {
		tmp = fma((((double) M_PI) * (a * a)), -0.011111111111111112, ((((double) M_PI) * (b * b)) * 0.011111111111111112)) * angle_m;
	} else {
		tmp = (((b * b) * 2.0) * (((double) M_PI) * (0.005555555555555556 * angle_m))) * fma((-1.54320987654321e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 1.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -2e+109)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	elseif (t_0 <= 2e+300)
		tmp = Float64(fma(Float64(pi * Float64(a * a)), -0.011111111111111112, Float64(Float64(pi * Float64(b * b)) * 0.011111111111111112)) * angle_m);
	else
		tmp = Float64(Float64(Float64(Float64(b * b) * 2.0) * Float64(pi * Float64(0.005555555555555556 * angle_m))) * fma(Float64(-1.54320987654321e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 1.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -2e+109], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(N[(N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112 + N[(N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -1.99999999999999996e109

   1. Initial program 52.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.4

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 50.3

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 50.5

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 50.5%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 65.2

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 65.2%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -1.99999999999999996e109 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.0000000000000001e300

   1. Initial program 62.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\frac{angle}{180}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

      13. lift-PI.f64 62.5

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \]

   3. Applied rewrites 62.5%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\pi}{2}\right)} \]

   4. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto 2 \cdot \left({b}^{2} \cdot \left(\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + \color{blue}{a \cdot \left(-2 \cdot \left(a \cdot \left(\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{-1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]

   9. Applied rewrites 62.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right), \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot b\right)\right) \cdot 2\right), \color{blue}{a}, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto angle \cdot \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{1}{90} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{90} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

   12. Applied rewrites 57.8%

       \[\leadsto \mathsf{fma}\left(\pi \cdot \left(a \cdot a\right), -0.011111111111111112, \left(\pi \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right) \cdot angle \]

   if 2.0000000000000001e300 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 40.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 37.9

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{\color{blue}{angle}}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \frac{angle}{\color{blue}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. mult-flip N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      13. lower-*.f64 37.9

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Applied rewrites 37.9%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \left(\left(\frac{-1}{64800} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot {angle}^{2}, {\color{blue}{\mathsf{PI}\left(\right)}}^{2}, 1\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 1\right) \]

      10. lift-PI.f64 39.6

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   7. Applied rewrites 39.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \left(\color{blue}{\left(2 \cdot {b}^{2}\right)} \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \color{blue}{2}\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \color{blue}{2}\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{64800} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

      4. lift-*.f64 49.2

         \[\leadsto \left(\left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]

   10. Applied rewrites 49.2%

       \[\leadsto \left(\color{blue}{\left(\left(b \cdot b\right) \cdot 2\right)} \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \mathsf{fma}\left(-1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{+58}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -5e+58)
    (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
    (* (* (* (* PI (+ a b)) (- b a)) angle_m) 0.011111111111111112))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -5e+58) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else {
		tmp = (((((double) M_PI) * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -5e+58) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else {
		tmp = (((Math.PI * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -5e+58:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	else:
		tmp = (((math.pi * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -5e+58)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	else
		tmp = Float64(Float64(Float64(Float64(pi * Float64(a + b)) * Float64(b - a)) * angle_m) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -5e+58)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	else
		tmp = (((pi * (a + b)) * (b - a)) * angle_m) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+58], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $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)))) < -4.99999999999999986e58

   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. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 51.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 51.0

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 51.0%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.2

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.2%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 64.9

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4.99999999999999986e58 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\frac{angle}{180}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(\frac{1}{180} \cdot angle\right)\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \]

      13. lift-PI.f64 53.8

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \]

   3. Applied rewrites 53.8%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\left(-\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) + \frac{\pi}{2}\right)} \]

   4. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 2\right) \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot angle\right) \cdot 0.011111111111111112} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 57.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e-7)
    (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
    (* (* (* 0.011111111111111112 angle_m) PI) (* (+ b a) (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -4e-7) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else {
		tmp = ((0.011111111111111112 * angle_m) * ((double) M_PI)) * ((b + a) * (b - a));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -4e-7) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else {
		tmp = ((0.011111111111111112 * angle_m) * Math.PI) * ((b + a) * (b - a));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -4e-7:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	else:
		tmp = ((0.011111111111111112 * angle_m) * math.pi) * ((b + a) * (b - a))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -4e-7)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * pi) * Float64(Float64(b + a) * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -4e-7)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	else
		tmp = ((0.011111111111111112 * angle_m) * pi) * ((b + a) * (b - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-7], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b + a), $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)))) < -3.9999999999999998e-7

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 51.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 50.8

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.8%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 51.0

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 51.0%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 63.6

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 63.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -3.9999999999999998e-7 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 50.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\pi} \cdot \left(b \cdot b - a \cdot a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - \color{blue}{a} \cdot a\right)\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      9. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right) \]

      10. pow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right) \]

      11. pow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      13. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left({\color{blue}{b}}^{2} - {a}^{2}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left({b}^{\color{blue}{2}} - {a}^{2}\right) \]

      17. pow2 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left({b}^{2} - a \cdot \color{blue}{a}\right) \]

      18. pow2 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b - \color{blue}{a} \cdot a\right) \]

      19. difference-of-squares N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

      21. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right) \]

      22. lower--.f64 55.6

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right) \]

   6. Applied rewrites 55.6%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 57.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -5 \cdot 10^{-258}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -5e-258)
    (* (* -0.011111111111111112 a) (* (* angle_m PI) a))
    (* (* (* PI (* b b)) angle_m) 0.011111111111111112))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -5e-258) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	} else {
		tmp = ((((double) M_PI) * (b * b)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -5e-258) {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	} else {
		tmp = ((Math.PI * (b * b)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -5e-258:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	else:
		tmp = ((math.pi * (b * b)) * angle_m) * 0.011111111111111112
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -5e-258)
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	else
		tmp = Float64(Float64(Float64(pi * Float64(b * b)) * angle_m) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -5e-258)
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	else
		tmp = ((pi * (b * b)) * angle_m) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-258], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $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)))) < -4.9999999999999999e-258

   1. Initial program 54.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 51.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 50.8

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 50.8%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 50.9

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 50.9%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 61.2

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 61.2%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]

   if -4.9999999999999999e-258 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 49.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      8. pow2 N/A

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      9. lift-*.f64 54.1

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 54.1%

      \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \color{blue}{0.011111111111111112} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 38.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{-170}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.05e-170)
    (* (* -0.011111111111111112 (* a a)) (* PI angle_m))
    (* (* -0.011111111111111112 a) (* (* angle_m PI) a)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.05e-170) {
		tmp = (-0.011111111111111112 * (a * a)) * (((double) M_PI) * angle_m);
	} else {
		tmp = (-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.05e-170) {
		tmp = (-0.011111111111111112 * (a * a)) * (Math.PI * angle_m);
	} else {
		tmp = (-0.011111111111111112 * a) * ((angle_m * Math.PI) * a);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 1.05e-170:
		tmp = (-0.011111111111111112 * (a * a)) * (math.pi * angle_m)
	else:
		tmp = (-0.011111111111111112 * a) * ((angle_m * math.pi) * a)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 1.05e-170)
		tmp = Float64(Float64(-0.011111111111111112 * Float64(a * a)) * Float64(pi * angle_m));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.05e-170)
		tmp = (-0.011111111111111112 * (a * a)) * (pi * angle_m);
	else
		tmp = (-0.011111111111111112 * a) * ((angle_m * pi) * a);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.05e-170], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.05e-170

   1. Initial program 57.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 53.5

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 35.4

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 35.4%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]

   if 1.05e-170 < a

   1. Initial program 49.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 45.3

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 34.1

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 34.1%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 34.2

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 34.2%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

       13. lift-PI.f64 40.5

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

   11. Applied rewrites 40.5%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 37.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* -0.011111111111111112 a) (* (* angle_m PI) a))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((-0.011111111111111112 * a) * ((angle_m * ((double) M_PI)) * a));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((-0.011111111111111112 * a) * ((angle_m * Math.PI) * a));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((-0.011111111111111112 * a) * ((angle_m * math.pi) * a))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(-0.011111111111111112 * a) * Float64(Float64(angle_m * pi) * a)))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((-0.011111111111111112 * a) * ((angle_m * pi) * a));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   10. lower-*.f64 50.3

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

5. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   4. pow2 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   8. lift-PI.f64 34.9

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

7. Applied rewrites 34.9%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   5. lower-*.f64 35.0

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

9. Applied rewrites 35.0%

   \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    4. lift-PI.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

    6. associate-*l* N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right) \]

    7. *-commutative N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \]

    13. lift-PI.f64 38.5

        \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right) \]

11. Applied rewrites 38.5%

    \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{a}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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)))))