ab-angle->ABCF B

Percentage Accurate: 54.1% → 65.6%

Time: 6.3s

Alternatives: 16

Speedup: 5.5×

• 33.5% 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: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_1 := -t\_0\\ t_2 := \sin t\_0 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, t\_2, 0\right), b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot t\_2\right)\\ \mathbf{elif}\;a\_m \leq 1.95 \cdot 10^{+250}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 2\right) \cdot \frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle\_m, t\_1\right)\right)}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556))
        (t_1 (- t_0))
        (t_2
         (*
          (sin t_0)
          (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))))))
   (*
    angle_s
    (if (<= a_m 8e+162)
      (fma (* 2.0 (fma b t_2 0.0)) b (* (* -2.0 (* a_m a_m)) t_2))
      (if (<= a_m 1.95e+250)
        (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
        (*
         (* (* (+ b a_m) (- b a_m)) 2.0)
         (/
          (+
           (sin (- t_0 t_1))
           (sin (fma (* PI 0.005555555555555556) angle_m t_1)))
          2.0)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_1 = -t_0;
	double t_2 = sin(t_0) * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)));
	double tmp;
	if (a_m <= 8e+162) {
		tmp = fma((2.0 * fma(b, t_2, 0.0)), b, ((-2.0 * (a_m * a_m)) * t_2));
	} else if (a_m <= 1.95e+250) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else {
		tmp = (((b + a_m) * (b - a_m)) * 2.0) * ((sin((t_0 - t_1)) + sin(fma((((double) M_PI) * 0.005555555555555556), angle_m, t_1))) / 2.0);
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_1 = Float64(-t_0)
	t_2 = Float64(sin(t_0) * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0))))
	tmp = 0.0
	if (a_m <= 8e+162)
		tmp = fma(Float64(2.0 * fma(b, t_2, 0.0)), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * t_2));
	elseif (a_m <= 1.95e+250)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	else
		tmp = Float64(Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * 2.0) * Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(pi * 0.005555555555555556), angle_m, t_1))) / 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a$95$m, 8e+162], N[(N[(2.0 * N[(b * t$95$2 + 0.0), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.95e+250], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.9999999999999995e162

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in b around 0

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

   6. Applied rewrites 60.7%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 60.8

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

   8. Applied rewrites 60.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 60.8

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

   10. Applied rewrites 60.8%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lift-PI.f64 61.0

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

   12. Applied rewrites 61.0%

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

   13. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right), 0\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right)\right) \]
   14. Step-by-step derivation

   15. Applied rewrites 61.0%

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

   if 7.9999999999999995e162 < a < 1.95e250

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if 1.95e250 < a

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

   5. Applied rewrites 57.8%

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

Alternative 2: 65.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_1 := \sin t\_0\\ t_2 := t\_1 \cdot \cos t\_0\\ t_3 := t\_1 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a\_m, t\_2, \left(t\_2 \cdot \left(0 \cdot b\right)\right) \cdot 2\right), a\_m, \left(\left(b \cdot b\right) \cdot 2\right) \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, t\_3, 0\right), b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot t\_3\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556))
        (t_1 (sin t_0))
        (t_2 (* t_1 (cos t_0)))
        (t_3
         (* t_1 (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))))))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -4e+20)
      (fma
       (fma (* -2.0 a_m) t_2 (* (* t_2 (* 0.0 b)) 2.0))
       a_m
       (* (* (* b b) 2.0) t_2))
      (fma (* 2.0 (fma b t_3 0.0)) b (* (* -2.0 (* a_m a_m)) t_3))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_1 = sin(t_0);
	double t_2 = t_1 * cos(t_0);
	double t_3 = t_1 * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)));
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -4e+20) {
		tmp = fma(fma((-2.0 * a_m), t_2, ((t_2 * (0.0 * b)) * 2.0)), a_m, (((b * b) * 2.0) * t_2));
	} else {
		tmp = fma((2.0 * fma(b, t_3, 0.0)), b, ((-2.0 * (a_m * a_m)) * t_3));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_1 = sin(t_0)
	t_2 = Float64(t_1 * cos(t_0))
	t_3 = Float64(t_1 * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0))))
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -4e+20)
		tmp = fma(fma(Float64(-2.0 * a_m), t_2, Float64(Float64(t_2 * Float64(0.0 * b)) * 2.0)), a_m, Float64(Float64(Float64(b * b) * 2.0) * t_2));
	else
		tmp = fma(Float64(2.0 * fma(b, t_3, 0.0)), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * t_3));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+20], N[(N[(N[(-2.0 * a$95$m), $MachinePrecision] * t$95$2 + N[(N[(t$95$2 * N[(0.0 * b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * a$95$m + N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b * t$95$3 + 0.0), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * t$95$3), $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)))) < -4e20

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 60.7%

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

   if -4e20 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in b around 0

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

   6. Applied rewrites 60.7%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 60.8

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

   8. Applied rewrites 60.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 60.8

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

   10. Applied rewrites 60.8%

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

       1. lift-cos.f64 N/A

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

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

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

       3. lower-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lift-PI.f64 61.0

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

   12. Applied rewrites 61.0%

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

   13. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right), 0\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right)\right) \]
   14. Step-by-step derivation

   15. Applied rewrites 61.0%

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right), 0\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 65.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi \cdot angle\_m}{180}\\ t_1 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ t_2 := \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;t\_2 \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \left(a\_m \cdot a\_m\right)\right) \cdot -0.011111111111111112, angle\_m, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle\_m, 0 \cdot angle\_m\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot angle\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (/ (* PI angle_m) 180.0))
        (t_1 (* 2.0 (- (pow b 2.0) (pow a_m 2.0))))
        (t_2 (* (* (+ b a_m) (- b a_m)) 2.0)))
   (*
    angle_s
    (if (<= t_1 (- INFINITY))
      (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
      (if (<= t_1 2e+291)
        (* t_2 (* (sin t_0) (cos t_0)))
        (if (<= t_1 INFINITY)
          (fma
           (* (* PI (* a_m a_m)) -0.011111111111111112)
           angle_m
           (*
            (* 0.011111111111111112 (fma (* PI b) angle_m (* 0.0 angle_m)))
            b))
          (*
           t_2
           (*
            (fma
             PI
             0.005555555555555556
             (*
              (* (* (* PI PI) PI) -1.1431184270690443e-7)
              (* angle_m angle_m)))
            angle_m))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, 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_m, 2.0));
	double t_2 = ((b + a_m) * (b - a_m)) * 2.0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else if (t_1 <= 2e+291) {
		tmp = t_2 * (sin(t_0) * cos(t_0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(((((double) M_PI) * (a_m * a_m)) * -0.011111111111111112), angle_m, ((0.011111111111111112 * fma((((double) M_PI) * b), angle_m, (0.0 * angle_m))) * b));
	} else {
		tmp = t_2 * (fma(((double) M_PI), 0.005555555555555556, ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7) * (angle_m * angle_m))) * angle_m);
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) / 180.0)
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	t_2 = Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * 2.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	elseif (t_1 <= 2e+291)
		tmp = Float64(t_2 * Float64(sin(t_0) * cos(t_0)));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(Float64(pi * Float64(a_m * a_m)) * -0.011111111111111112), angle_m, Float64(Float64(0.011111111111111112 * fma(Float64(pi * b), angle_m, Float64(0.0 * angle_m))) * b));
	else
		tmp = Float64(t_2 * Float64(fma(pi, 0.005555555555555556, Float64(Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7) * Float64(angle_m * angle_m))) * angle_m));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] / 180.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+291], N[(t$95$2 * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(Pi * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * angle$95$m + N[(N[(0.011111111111111112 * N[(N[(Pi * b), $MachinePrecision] * angle$95$m + N[(0.0 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(Pi * 0.005555555555555556 + N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 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 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.9999999999999999e291

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 57.6

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

   5. Applied rewrites 57.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. mult-flip N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 57.8

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

   7. Applied rewrites 57.8%

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

   if 1.9999999999999999e291 < (*.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 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 53.8%

       \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \left(a \cdot a\right)\right) \cdot -0.011111111111111112, \color{blue}{angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, 0 \cdot angle\right)\right) \cdot b\right) \]

   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 54.1%

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{angle}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 52.6%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ t_1 := \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 2\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+295}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;t\_1 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot \cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \left(a\_m \cdot a\_m\right)\right) \cdot -0.011111111111111112, angle\_m, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle\_m, 0 \cdot angle\_m\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot angle\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a_m 2.0))))
        (t_1 (* (* (+ b a_m) (- b a_m)) 2.0)))
   (*
    angle_s
    (if (<= t_0 -5e+295)
      (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
      (if (<= t_0 5e+294)
        (*
         t_1
         (*
          (sin (* PI (* 0.005555555555555556 angle_m)))
          (cos (* (* PI angle_m) 0.005555555555555556))))
        (if (<= t_0 INFINITY)
          (fma
           (* (* PI (* a_m a_m)) -0.011111111111111112)
           angle_m
           (*
            (* 0.011111111111111112 (fma (* PI b) angle_m (* 0.0 angle_m)))
            b))
          (*
           t_1
           (*
            (fma
             PI
             0.005555555555555556
             (*
              (* (* (* PI PI) PI) -1.1431184270690443e-7)
              (* angle_m angle_m)))
            angle_m))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a_m, 2.0));
	double t_1 = ((b + a_m) * (b - a_m)) * 2.0;
	double tmp;
	if (t_0 <= -5e+295) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else if (t_0 <= 5e+294) {
		tmp = t_1 * (sin((((double) M_PI) * (0.005555555555555556 * angle_m))) * cos(((((double) M_PI) * angle_m) * 0.005555555555555556)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(((((double) M_PI) * (a_m * a_m)) * -0.011111111111111112), angle_m, ((0.011111111111111112 * fma((((double) M_PI) * b), angle_m, (0.0 * angle_m))) * b));
	} else {
		tmp = t_1 * (fma(((double) M_PI), 0.005555555555555556, ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7) * (angle_m * angle_m))) * angle_m);
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	t_1 = Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * 2.0)
	tmp = 0.0
	if (t_0 <= -5e+295)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	elseif (t_0 <= 5e+294)
		tmp = Float64(t_1 * Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle_m))) * cos(Float64(Float64(pi * angle_m) * 0.005555555555555556))));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(Float64(pi * Float64(a_m * a_m)) * -0.011111111111111112), angle_m, Float64(Float64(0.011111111111111112 * fma(Float64(pi * b), angle_m, Float64(0.0 * angle_m))) * b));
	else
		tmp = Float64(t_1 * Float64(fma(pi, 0.005555555555555556, Float64(Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7) * Float64(angle_m * angle_m))) * angle_m));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -5e+295], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+294], N[(t$95$1 * N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(Pi * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * angle$95$m + N[(N[(0.011111111111111112 * N[(N[(Pi * b), $MachinePrecision] * angle$95$m + N[(0.0 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(Pi * 0.005555555555555556 + N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.99999999999999991e295

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if -4.99999999999999991e295 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 4.9999999999999999e294

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if 4.9999999999999999e294 < (*.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 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 53.8%

       \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \left(a \cdot a\right)\right) \cdot -0.011111111111111112, \color{blue}{angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, 0 \cdot angle\right)\right) \cdot b\right) \]

   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 54.1%

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{angle}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 52.6%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+287}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot \sin \left(2 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \left(a\_m \cdot a\_m\right)\right) \cdot -0.011111111111111112, angle\_m, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle\_m, 0 \cdot angle\_m\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot angle\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a_m 2.0)))))
   (*
    angle_s
    (if (<= t_0 -1e+287)
      (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
      (if (<= t_0 2e+291)
        (*
         (* (- b a_m) (+ a_m b))
         (sin (* 2.0 (* (* PI angle_m) 0.005555555555555556))))
        (if (<= t_0 INFINITY)
          (fma
           (* (* PI (* a_m a_m)) -0.011111111111111112)
           angle_m
           (*
            (* 0.011111111111111112 (fma (* PI b) angle_m (* 0.0 angle_m)))
            b))
          (*
           (* (* (+ b a_m) (- b a_m)) 2.0)
           (*
            (fma
             PI
             0.005555555555555556
             (*
              (* (* (* PI PI) PI) -1.1431184270690443e-7)
              (* angle_m angle_m)))
            angle_m))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a_m, 2.0));
	double tmp;
	if (t_0 <= -1e+287) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else if (t_0 <= 2e+291) {
		tmp = ((b - a_m) * (a_m + b)) * sin((2.0 * ((((double) M_PI) * angle_m) * 0.005555555555555556)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(((((double) M_PI) * (a_m * a_m)) * -0.011111111111111112), angle_m, ((0.011111111111111112 * fma((((double) M_PI) * b), angle_m, (0.0 * angle_m))) * b));
	} else {
		tmp = (((b + a_m) * (b - a_m)) * 2.0) * (fma(((double) M_PI), 0.005555555555555556, ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7) * (angle_m * angle_m))) * angle_m);
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -1e+287)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	elseif (t_0 <= 2e+291)
		tmp = Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * sin(Float64(2.0 * Float64(Float64(pi * angle_m) * 0.005555555555555556))));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(Float64(pi * Float64(a_m * a_m)) * -0.011111111111111112), angle_m, Float64(Float64(0.011111111111111112 * fma(Float64(pi * b), angle_m, Float64(0.0 * angle_m))) * b));
	else
		tmp = Float64(Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * 2.0) * Float64(fma(pi, 0.005555555555555556, Float64(Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7) * Float64(angle_m * angle_m))) * angle_m));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -1e+287], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+291], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(Pi * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * angle$95$m + N[(N[(0.011111111111111112 * N[(N[(Pi * b), $MachinePrecision] * angle$95$m + N[(0.0 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(Pi * 0.005555555555555556 + N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -1.0000000000000001e287

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if -1.0000000000000001e287 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.9999999999999999e291

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

   5. Applied rewrites 57.8%

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

   if 1.9999999999999999e291 < (*.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 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 53.8%

       \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \left(a \cdot a\right)\right) \cdot -0.011111111111111112, \color{blue}{angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, 0 \cdot angle\right)\right) \cdot b\right) \]

   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 54.1%

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

   3. Applied rewrites 57.9%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{angle}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + {angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 52.6%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 61.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \left(a\_m \cdot a\_m\right)\right) \cdot -0.011111111111111112, angle\_m, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle\_m, 0 \cdot angle\_m\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot \sin \left(\mathsf{fma}\left(\pi \cdot 0.005555555555555556, angle\_m, \frac{\pi}{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 3.4e-12)
    (fma
     (* (* PI (* a_m a_m)) -0.011111111111111112)
     angle_m
     (* (* 0.011111111111111112 (fma (* PI b) angle_m (* 0.0 angle_m))) b))
    (*
     (* (* (+ b a_m) (- b a_m)) 2.0)
     (*
      (sin (* PI (* 0.005555555555555556 angle_m)))
      (sin (fma (* PI 0.005555555555555556) angle_m (/ PI 2.0))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.4e-12) {
		tmp = fma(((((double) M_PI) * (a_m * a_m)) * -0.011111111111111112), angle_m, ((0.011111111111111112 * fma((((double) M_PI) * b), angle_m, (0.0 * angle_m))) * b));
	} else {
		tmp = (((b + a_m) * (b - a_m)) * 2.0) * (sin((((double) M_PI) * (0.005555555555555556 * angle_m))) * sin(fma((((double) M_PI) * 0.005555555555555556), angle_m, (((double) M_PI) / 2.0))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 3.4e-12)
		tmp = fma(Float64(Float64(pi * Float64(a_m * a_m)) * -0.011111111111111112), angle_m, Float64(Float64(0.011111111111111112 * fma(Float64(pi * b), angle_m, Float64(0.0 * angle_m))) * b));
	else
		tmp = Float64(Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * 2.0) * Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle_m))) * sin(fma(Float64(pi * 0.005555555555555556), angle_m, Float64(pi / 2.0)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 3.4e-12], N[(N[(N[(Pi * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * angle$95$m + N[(N[(0.011111111111111112 * N[(N[(Pi * b), $MachinePrecision] * angle$95$m + N[(0.0 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 3.4000000000000001e-12

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 53.8%

       \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \left(a \cdot a\right)\right) \cdot -0.011111111111111112, \color{blue}{angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, 0 \cdot angle\right)\right) \cdot b\right) \]

   if 3.4000000000000001e-12 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. lower-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-PI.f64 57.6

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

   5. Applied rewrites 57.6%

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

Alternative 7: 60.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 7 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \left(a\_m \cdot a\_m\right)\right) \cdot -0.011111111111111112, angle\_m, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle\_m, 0 \cdot angle\_m\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 7e+153)
    (fma
     (* (* PI (* a_m a_m)) -0.011111111111111112)
     angle_m
     (* (* 0.011111111111111112 (fma (* PI b) angle_m (* 0.0 angle_m))) b))
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 7e+153) {
		tmp = fma(((((double) M_PI) * (a_m * a_m)) * -0.011111111111111112), angle_m, ((0.011111111111111112 * fma((((double) M_PI) * b), angle_m, (0.0 * angle_m))) * b));
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 7e+153)
		tmp = fma(Float64(Float64(pi * Float64(a_m * a_m)) * -0.011111111111111112), angle_m, Float64(Float64(0.011111111111111112 * fma(Float64(pi * b), angle_m, Float64(0.0 * angle_m))) * b));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 7e+153], N[(N[(N[(Pi * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * angle$95$m + N[(N[(0.011111111111111112 * N[(N[(Pi * b), $MachinePrecision] * angle$95$m + N[(0.0 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 6.9999999999999998e153

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 53.8%

       \[\leadsto \mathsf{fma}\left(\left(\pi \cdot \left(a \cdot a\right)\right) \cdot -0.011111111111111112, \color{blue}{angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, 0 \cdot angle\right)\right) \cdot b\right) \]

   if 6.9999999999999998e153 < a

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

Alternative 8: 58.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;\left(\left(\left(\pi \cdot \left(a\_m + b\right)\right) \cdot \left(b - a\_m\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 9.5e+152)
    (* (* (* (* PI (+ a_m b)) (- b a_m)) angle_m) 0.011111111111111112)
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = (((((double) M_PI) * (a_m + b)) * (b - a_m)) * angle_m) * 0.011111111111111112;
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = (((Math.PI * (a_m + b)) * (b - a_m)) * angle_m) * 0.011111111111111112;
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 9.5e+152:
		tmp = (((math.pi * (a_m + b)) * (b - a_m)) * angle_m) * 0.011111111111111112
	else:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 9.5e+152)
		tmp = Float64(Float64(Float64(Float64(pi * Float64(a_m + b)) * Float64(b - a_m)) * angle_m) * 0.011111111111111112);
	else
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 9.5e+152)
		tmp = (((pi * (a_m + b)) * (b - a_m)) * angle_m) * 0.011111111111111112;
	else
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 9.5e+152], N[(N[(N[(N[(Pi * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.49999999999999916e152

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 54.7%

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

   if 9.49999999999999916e152 < a

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

Alternative 9: 58.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;\left(\left(\pi \cdot \left(a\_m + b\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(0.011111111111111112 \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 9.5e+152)
    (* (* (* PI (+ a_m b)) (- b a_m)) (* 0.011111111111111112 angle_m))
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = ((((double) M_PI) * (a_m + b)) * (b - a_m)) * (0.011111111111111112 * angle_m);
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = ((Math.PI * (a_m + b)) * (b - a_m)) * (0.011111111111111112 * angle_m);
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 9.5e+152:
		tmp = ((math.pi * (a_m + b)) * (b - a_m)) * (0.011111111111111112 * angle_m)
	else:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 9.5e+152)
		tmp = Float64(Float64(Float64(pi * Float64(a_m + b)) * Float64(b - a_m)) * Float64(0.011111111111111112 * angle_m));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 9.5e+152)
		tmp = ((pi * (a_m + b)) * (b - a_m)) * (0.011111111111111112 * angle_m);
	else
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 9.5e+152], N[(N[(N[(Pi * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * angle$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.49999999999999916e152

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-*.f64 54.7

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

   6. Applied rewrites 54.7%

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

   if 9.49999999999999916e152 < a

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

Alternative 10: 58.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(a\_m + b\right) \cdot \left(b - a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 1.35e+154)
    (* (* (* angle_m PI) 0.011111111111111112) (* (+ a_m b) (- b a_m)))
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 1.35e+154) {
		tmp = ((angle_m * ((double) M_PI)) * 0.011111111111111112) * ((a_m + b) * (b - a_m));
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 1.35e+154) {
		tmp = ((angle_m * Math.PI) * 0.011111111111111112) * ((a_m + b) * (b - a_m));
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 1.35e+154:
		tmp = ((angle_m * math.pi) * 0.011111111111111112) * ((a_m + b) * (b - a_m))
	else:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 1.35e+154)
		tmp = Float64(Float64(Float64(angle_m * pi) * 0.011111111111111112) * Float64(Float64(a_m + b) * Float64(b - a_m)));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 1.35e+154)
		tmp = ((angle_m * pi) * 0.011111111111111112) * ((a_m + b) * (b - a_m));
	else
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 1.35e+154], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.35000000000000003e154

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-commutative N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lift-PI.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \log \left({\left(e^{\pi}\right)}^{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \log \left({\left(e^{\pi}\right)}^{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \log \left({\left(e^{\pi}\right)}^{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \]

      15. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \log \left({\left(e^{\pi}\right)}^{\left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \log \left({\left(e^{\pi}\right)}^{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \]

      17. lower-+.f64 36.1

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

   6. Applied rewrites 36.1%

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

   8. Applied rewrites 54.7%

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

   if 1.35000000000000003e154 < a

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

Alternative 11: 58.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 9.5e+152)
    (* (* (* 0.011111111111111112 angle_m) PI) (* (- b a_m) (+ a_m b)))
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = ((0.011111111111111112 * angle_m) * ((double) M_PI)) * ((b - a_m) * (a_m + b));
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = ((0.011111111111111112 * angle_m) * Math.PI) * ((b - a_m) * (a_m + b));
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 9.5e+152:
		tmp = ((0.011111111111111112 * angle_m) * math.pi) * ((b - a_m) * (a_m + b))
	else:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 9.5e+152)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * pi) * Float64(Float64(b - a_m) * Float64(a_m + b)));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 9.5e+152)
		tmp = ((0.011111111111111112 * angle_m) * pi) * ((b - a_m) * (a_m + b));
	else
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 9.5e+152], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.49999999999999916e152

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 54.8

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

   6. Applied rewrites 54.8%

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

   if 9.49999999999999916e152 < a

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

Alternative 12: 58.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 9.5e+152)
    (* (* 0.011111111111111112 angle_m) (* PI (* (+ b a_m) (- b a_m))))
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = (0.011111111111111112 * angle_m) * (((double) M_PI) * ((b + a_m) * (b - a_m)));
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 9.5e+152) {
		tmp = (0.011111111111111112 * angle_m) * (Math.PI * ((b + a_m) * (b - a_m)));
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 9.5e+152:
		tmp = (0.011111111111111112 * angle_m) * (math.pi * ((b + a_m) * (b - a_m)))
	else:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 9.5e+152)
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(pi * Float64(Float64(b + a_m) * Float64(b - a_m))));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 9.5e+152)
		tmp = (0.011111111111111112 * angle_m) * (pi * ((b + a_m) * (b - a_m)));
	else
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 9.5e+152], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(Pi * N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 9.49999999999999916e152

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   if 9.49999999999999916e152 < a

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

Alternative 13: 57.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -2 \cdot 10^{-247}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\pi \cdot \left(b \cdot \left(b - a\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -2e-247)
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
    (* (* 0.011111111111111112 angle_m) (* PI (* b (- b a_m)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -2e-247) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (((double) M_PI) * (b * (b - a_m)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -2e-247) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (Math.PI * (b * (b - a_m)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -2e-247:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	else:
		tmp = (0.011111111111111112 * angle_m) * (math.pi * (b * (b - a_m)))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -2e-247)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(pi * Float64(b * Float64(b - a_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -2e-247)
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	else
		tmp = (0.011111111111111112 * angle_m) * (pi * (b * (b - a_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-247], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(Pi * N[(b * N[(b - a$95$m), $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)))) < -2e-247

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if -2e-247 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 37.5%

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

Alternative 14: 57.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -2 \cdot 10^{-247}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -2e-247)
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
    (* (* (* PI (* b b)) angle_m) 0.011111111111111112))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -2e-247) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else {
		tmp = ((((double) M_PI) * (b * b)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -2e-247) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	} else {
		tmp = ((Math.PI * (b * b)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -2e-247:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	else:
		tmp = ((math.pi * (b * b)) * angle_m) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -2e-247)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	else
		tmp = Float64(Float64(Float64(pi * Float64(b * b)) * angle_m) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -2e-247)
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	else
		tmp = ((pi * (b * b)) * angle_m) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-247], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $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)))) < -2e-247

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

   if -2e-247 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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 34.7

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

   7. Applied rewrites 34.7%

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

Alternative 15: 40.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a\_m \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(\pi \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a_m 8.8e-169)
    (* (* -0.011111111111111112 (* a_m a_m)) (* PI angle_m))
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 8.8e-169) {
		tmp = (-0.011111111111111112 * (a_m * a_m)) * (((double) M_PI) * angle_m);
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (a_m <= 8.8e-169) {
		tmp = (-0.011111111111111112 * (a_m * a_m)) * (Math.PI * angle_m);
	} else {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if a_m <= 8.8e-169:
		tmp = (-0.011111111111111112 * (a_m * a_m)) * (math.pi * angle_m)
	else:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (a_m <= 8.8e-169)
		tmp = Float64(Float64(-0.011111111111111112 * Float64(a_m * a_m)) * Float64(pi * angle_m));
	else
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (a_m <= 8.8e-169)
		tmp = (-0.011111111111111112 * (a_m * a_m)) * (pi * angle_m);
	else
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a$95$m, 8.8e-169], N[(N[(-0.011111111111111112 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 8.80000000000000029e-169

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

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

   if 8.80000000000000029e-169 < a

   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 54.7

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

   4. Applied rewrites 54.7%

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

   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. lift-*.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 35.6

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

   7. Applied rewrites 35.6%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   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 35.7

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

   9. Applied rewrites 35.7%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   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. associate-*l* N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-*.f64 N/A

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

       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. lower-*.f64 N/A

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

       13. lift-PI.f64 38.9

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

   11. Applied rewrites 38.9%

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

Alternative 16: 38.9% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $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. unpow2 N/A

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

   7. unpow2 N/A

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

   8. difference-of-squares N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 54.7

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

4. Applied rewrites 54.7%

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

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. lift-*.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 35.6

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

7. Applied rewrites 35.6%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
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 35.7

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

9. Applied rewrites 35.7%

   \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
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. associate-*l* N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-*.f64 N/A

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

    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. lower-*.f64 N/A

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

    13. lift-PI.f64 38.9

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

11. Applied rewrites 38.9%

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

Reproduce

herbie shell --seed 2025132 
(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)))))