ab-angle->ABCF B

Percentage Accurate: 53.1% → 67.3%

Time: 7.5s

Alternatives: 9

Speedup: N/A×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.1% 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: 67.3% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_1 := \sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right)\\ t_2 := \sin t\_0\\ t_3 := \left(t\_2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.32 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, \cos \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right), t\_3 \cdot \cos \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_2 \cdot b, t\_1, \left(\left(0 \cdot a\right) \cdot t\_1\right) \cdot t\_2\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556))
        (t_1 (sin (fma 0.5 PI t_0)))
        (t_2 (sin t_0))
        (t_3 (* (* t_2 (+ a b)) (- b a))))
   (*
    angle_s
    (if (<= angle_m 1.32e+259)
      (fma
       t_3
       (cos (* 0.005555555555555556 (* angle_m PI)))
       (* t_3 (cos (* (* angle_m PI) 0.005555555555555556))))
      (fma
       (* 2.0 (fma (* t_2 b) t_1 (* (* (* 0.0 a) t_1) t_2)))
       b
       (* (* -2.0 (* a a)) (* t_1 t_2)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_1 = sin(fma(0.5, ((double) M_PI), t_0));
	double t_2 = sin(t_0);
	double t_3 = (t_2 * (a + b)) * (b - a);
	double tmp;
	if (angle_m <= 1.32e+259) {
		tmp = fma(t_3, cos((0.005555555555555556 * (angle_m * ((double) M_PI)))), (t_3 * cos(((angle_m * ((double) M_PI)) * 0.005555555555555556))));
	} else {
		tmp = fma((2.0 * fma((t_2 * b), t_1, (((0.0 * a) * t_1) * t_2))), b, ((-2.0 * (a * a)) * (t_1 * t_2)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_1 = sin(fma(0.5, pi, t_0))
	t_2 = sin(t_0)
	t_3 = Float64(Float64(t_2 * Float64(a + b)) * Float64(b - a))
	tmp = 0.0
	if (angle_m <= 1.32e+259)
		tmp = fma(t_3, cos(Float64(0.005555555555555556 * Float64(angle_m * pi))), Float64(t_3 * cos(Float64(Float64(angle_m * pi) * 0.005555555555555556))));
	else
		tmp = fma(Float64(2.0 * fma(Float64(t_2 * b), t_1, Float64(Float64(Float64(0.0 * a) * t_1) * t_2))), b, Float64(Float64(-2.0 * Float64(a * a)) * Float64(t_1 * t_2)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 1.32e+259], N[(t$95$3 * N[Cos[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$3 * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(t$95$2 * b), $MachinePrecision] * t$95$1 + N[(N[(N[(0.0 * a), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.31999999999999998e259

   1. Initial program 55.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in angle around inf

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 70.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sin-+PI/2 N/A

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

      8. associate-*r* N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 66.8

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

   10. Applied rewrites 66.8%

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

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. sin-+PI/2 N/A

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

       8. associate-*r* N/A

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

       9. lower-cos.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-PI.f64 71.9

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

   12. Applied rewrites 71.9%

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

   if 1.31999999999999998e259 < angle

   1. Initial program 20.4%

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

   4. Applied rewrites 44.5%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.1% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_1 := \sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right)\\ t_2 := \sin t\_0\\ t_3 := \left(\left(0 \cdot a\right) \cdot t\_1\right) \cdot t\_2\\ t_4 := \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(t\_1 \cdot t\_2\right)\\ t_5 := t\_2 \cdot b\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;angle\_m \leq 5.6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_5, \sin \left(0.5 \cdot \pi\right), t\_3\right), b, t\_4\right)\\ \mathbf{elif}\;angle\_m \leq 2.5 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot \cos t\_0\right) \cdot t\_2}{b}, \frac{-2}{b}, \sin \left(2 \cdot t\_0\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_5, t\_1, t\_3\right), b, t\_4\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556))
        (t_1 (sin (fma 0.5 PI t_0)))
        (t_2 (sin t_0))
        (t_3 (* (* (* 0.0 a) t_1) t_2))
        (t_4 (* (* -2.0 (* a a)) (* t_1 t_2)))
        (t_5 (* t_2 b)))
   (*
    angle_s
    (if (<= angle_m 2.6e-25)
      (* (* (* 0.011111111111111112 angle_m) (* PI (+ a b))) (- b a))
      (if (<= angle_m 5.6e+153)
        (fma (* 2.0 (fma t_5 (sin (* 0.5 PI)) t_3)) b t_4)
        (if (<= angle_m 2.5e+258)
          (*
           (fma
            (/ (* (* (* a a) (cos t_0)) t_2) b)
            (/ -2.0 b)
            (sin (* 2.0 t_0)))
           (* b b))
          (fma (* 2.0 (fma t_5 t_1 t_3)) b t_4)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_1 = sin(fma(0.5, ((double) M_PI), t_0));
	double t_2 = sin(t_0);
	double t_3 = ((0.0 * a) * t_1) * t_2;
	double t_4 = (-2.0 * (a * a)) * (t_1 * t_2);
	double t_5 = t_2 * b;
	double tmp;
	if (angle_m <= 2.6e-25) {
		tmp = ((0.011111111111111112 * angle_m) * (((double) M_PI) * (a + b))) * (b - a);
	} else if (angle_m <= 5.6e+153) {
		tmp = fma((2.0 * fma(t_5, sin((0.5 * ((double) M_PI))), t_3)), b, t_4);
	} else if (angle_m <= 2.5e+258) {
		tmp = fma(((((a * a) * cos(t_0)) * t_2) / b), (-2.0 / b), sin((2.0 * t_0))) * (b * b);
	} else {
		tmp = fma((2.0 * fma(t_5, t_1, t_3)), b, t_4);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_1 = sin(fma(0.5, pi, t_0))
	t_2 = sin(t_0)
	t_3 = Float64(Float64(Float64(0.0 * a) * t_1) * t_2)
	t_4 = Float64(Float64(-2.0 * Float64(a * a)) * Float64(t_1 * t_2))
	t_5 = Float64(t_2 * b)
	tmp = 0.0
	if (angle_m <= 2.6e-25)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(pi * Float64(a + b))) * Float64(b - a));
	elseif (angle_m <= 5.6e+153)
		tmp = fma(Float64(2.0 * fma(t_5, sin(Float64(0.5 * pi)), t_3)), b, t_4);
	elseif (angle_m <= 2.5e+258)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(a * a) * cos(t_0)) * t_2) / b), Float64(-2.0 / b), sin(Float64(2.0 * t_0))) * Float64(b * b));
	else
		tmp = fma(Float64(2.0 * fma(t_5, t_1, t_3)), b, t_4);
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(0.0 * a), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * b), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.6e-25], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 5.6e+153], N[(N[(2.0 * N[(t$95$5 * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] * b + t$95$4), $MachinePrecision], If[LessEqual[angle$95$m, 2.5e+258], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / b), $MachinePrecision] * N[(-2.0 / b), $MachinePrecision] + N[Sin[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$5 * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision] * b + t$95$4), $MachinePrecision]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if angle < 2.6e-25

   1. Initial program 59.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 60.1

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

   8. Applied rewrites 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 72.9

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

   10. Applied rewrites 72.9%

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

   if 2.6e-25 < angle < 5.5999999999999997e153

   1. Initial program 45.7%

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lift-PI.f64 43.2

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   10. Applied rewrites 43.2%

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   if 5.5999999999999997e153 < angle < 2.5e258

   1. Initial program 39.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 27.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

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

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-*.f64 43.7

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

   7. Applied rewrites 43.7%

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

   if 2.5e258 < angle

   1. Initial program 20.4%

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

   4. Applied rewrites 44.5%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.4% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := -2 \cdot \left(a \cdot a\right)\\ t_1 := \sin \left(0.5 \cdot \pi\right)\\ t_2 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_3 := \sin \left(\mathsf{fma}\left(0.5, \pi, t\_2\right)\right)\\ t_4 := \sin t\_2\\ t_5 := t\_4 \cdot b\\ t_6 := \left(\left(0 \cdot a\right) \cdot t\_3\right) \cdot t\_4\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 6.6 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right), \cos \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right), \left(\left(t\_4 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_5, t\_1, t\_6\right), b, t\_0 \cdot \left(t\_3 \cdot t\_4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_5, t\_3, t\_6\right), b, t\_0 \cdot \left(\mathsf{fma}\left(t\_1, \cos t\_2, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot t\_4\right) \cdot t\_4\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* a a)))
        (t_1 (sin (* 0.5 PI)))
        (t_2 (* (* PI angle_m) 0.005555555555555556))
        (t_3 (sin (fma 0.5 PI t_2)))
        (t_4 (sin t_2))
        (t_5 (* t_4 b))
        (t_6 (* (* (* 0.0 a) t_3) t_4)))
   (*
    angle_s
    (if (<= angle_m 6.6e+32)
      (fma
       (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b)) (- b a))
       (cos (* 0.005555555555555556 (* angle_m PI)))
       (*
        (* (* t_4 (+ a b)) (- b a))
        (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0)))))
      (if (<= angle_m 6e+153)
        (fma (* 2.0 (fma t_5 t_1 t_6)) b (* t_0 (* t_3 t_4)))
        (fma
         (* 2.0 (fma t_5 t_3 t_6))
         b
         (*
          t_0
          (*
           (fma t_1 (cos t_2) (* (sin (fma 0.5 PI (/ PI 2.0))) t_4))
           t_4))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = -2.0 * (a * a);
	double t_1 = sin((0.5 * ((double) M_PI)));
	double t_2 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_3 = sin(fma(0.5, ((double) M_PI), t_2));
	double t_4 = sin(t_2);
	double t_5 = t_4 * b;
	double t_6 = ((0.0 * a) * t_3) * t_4;
	double tmp;
	if (angle_m <= 6.6e+32) {
		tmp = fma(((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b)) * (b - a)), cos((0.005555555555555556 * (angle_m * ((double) M_PI)))), (((t_4 * (a + b)) * (b - a)) * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)))));
	} else if (angle_m <= 6e+153) {
		tmp = fma((2.0 * fma(t_5, t_1, t_6)), b, (t_0 * (t_3 * t_4)));
	} else {
		tmp = fma((2.0 * fma(t_5, t_3, t_6)), b, (t_0 * (fma(t_1, cos(t_2), (sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))) * t_4)) * t_4)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(-2.0 * Float64(a * a))
	t_1 = sin(Float64(0.5 * pi))
	t_2 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_3 = sin(fma(0.5, pi, t_2))
	t_4 = sin(t_2)
	t_5 = Float64(t_4 * b)
	t_6 = Float64(Float64(Float64(0.0 * a) * t_3) * t_4)
	tmp = 0.0
	if (angle_m <= 6.6e+32)
		tmp = fma(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b)) * Float64(b - a)), cos(Float64(0.005555555555555556 * Float64(angle_m * pi))), Float64(Float64(Float64(t_4 * Float64(a + b)) * Float64(b - a)) * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))));
	elseif (angle_m <= 6e+153)
		tmp = fma(Float64(2.0 * fma(t_5, t_1, t_6)), b, Float64(t_0 * Float64(t_3 * t_4)));
	else
		tmp = fma(Float64(2.0 * fma(t_5, t_3, t_6)), b, Float64(t_0 * Float64(fma(t_1, cos(t_2), Float64(sin(fma(0.5, pi, Float64(pi / 2.0))) * t_4)) * t_4)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * b), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(0.0 * a), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$4), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 6.6e+32], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(t$95$4 * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 6e+153], N[(N[(2.0 * N[(t$95$5 * t$95$1 + t$95$6), $MachinePrecision]), $MachinePrecision] * b + N[(t$95$0 * N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$5 * t$95$3 + t$95$6), $MachinePrecision]), $MachinePrecision] * b + N[(t$95$0 * N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision] + N[(N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 6.60000000000000039e32

   1. Initial program 60.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in angle around inf

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

      1. count-2-rev N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 80.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sin-+PI/2 N/A

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

      8. associate-*r* N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 76.9

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

   10. Applied rewrites 76.9%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 78.4

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

   12. Applied rewrites 78.4%

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

   if 6.60000000000000039e32 < angle < 6.00000000000000037e153

   1. Initial program 35.9%

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

   4. Applied rewrites 28.7%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lift-PI.f64 32.3

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   10. Applied rewrites 32.3%

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   if 6.00000000000000037e153 < angle

   1. Initial program 32.1%

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 45.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 39.0%

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(0.5 \cdot \pi\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 65.4% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \pi\right)\\ t_1 := -2 \cdot \left(a \cdot a\right)\\ t_2 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_3 := \sin \left(\mathsf{fma}\left(0.5, \pi, t\_2\right)\right)\\ t_4 := \sin t\_2\\ t_5 := \left(\left(0 \cdot a\right) \cdot t\_3\right) \cdot t\_4\\ t_6 := t\_4 \cdot b\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;angle\_m \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_6, t\_0, t\_5\right), b, t\_1 \cdot \left(t\_3 \cdot t\_4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_6, t\_3, t\_5\right), b, t\_1 \cdot \left(\mathsf{fma}\left(t\_0, \cos t\_2, \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot t\_4\right) \cdot t\_4\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 PI)))
        (t_1 (* -2.0 (* a a)))
        (t_2 (* (* PI angle_m) 0.005555555555555556))
        (t_3 (sin (fma 0.5 PI t_2)))
        (t_4 (sin t_2))
        (t_5 (* (* (* 0.0 a) t_3) t_4))
        (t_6 (* t_4 b)))
   (*
    angle_s
    (if (<= angle_m 2.6e-25)
      (* (* (* 0.011111111111111112 angle_m) (* PI (+ a b))) (- b a))
      (if (<= angle_m 6e+153)
        (fma (* 2.0 (fma t_6 t_0 t_5)) b (* t_1 (* t_3 t_4)))
        (fma
         (* 2.0 (fma t_6 t_3 t_5))
         b
         (*
          t_1
          (*
           (fma t_0 (cos t_2) (* (sin (fma 0.5 PI (/ PI 2.0))) t_4))
           t_4))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((0.5 * ((double) M_PI)));
	double t_1 = -2.0 * (a * a);
	double t_2 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_3 = sin(fma(0.5, ((double) M_PI), t_2));
	double t_4 = sin(t_2);
	double t_5 = ((0.0 * a) * t_3) * t_4;
	double t_6 = t_4 * b;
	double tmp;
	if (angle_m <= 2.6e-25) {
		tmp = ((0.011111111111111112 * angle_m) * (((double) M_PI) * (a + b))) * (b - a);
	} else if (angle_m <= 6e+153) {
		tmp = fma((2.0 * fma(t_6, t_0, t_5)), b, (t_1 * (t_3 * t_4)));
	} else {
		tmp = fma((2.0 * fma(t_6, t_3, t_5)), b, (t_1 * (fma(t_0, cos(t_2), (sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))) * t_4)) * t_4)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(0.5 * pi))
	t_1 = Float64(-2.0 * Float64(a * a))
	t_2 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_3 = sin(fma(0.5, pi, t_2))
	t_4 = sin(t_2)
	t_5 = Float64(Float64(Float64(0.0 * a) * t_3) * t_4)
	t_6 = Float64(t_4 * b)
	tmp = 0.0
	if (angle_m <= 2.6e-25)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(pi * Float64(a + b))) * Float64(b - a));
	elseif (angle_m <= 6e+153)
		tmp = fma(Float64(2.0 * fma(t_6, t_0, t_5)), b, Float64(t_1 * Float64(t_3 * t_4)));
	else
		tmp = fma(Float64(2.0 * fma(t_6, t_3, t_5)), b, Float64(t_1 * Float64(fma(t_0, cos(t_2), Float64(sin(fma(0.5, pi, Float64(pi / 2.0))) * t_4)) * t_4)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(0.0 * a), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * b), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.6e-25], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 6e+153], N[(N[(2.0 * N[(t$95$6 * t$95$0 + t$95$5), $MachinePrecision]), $MachinePrecision] * b + N[(t$95$1 * N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$6 * t$95$3 + t$95$5), $MachinePrecision]), $MachinePrecision] * b + N[(t$95$1 * N[(N[(t$95$0 * N[Cos[t$95$2], $MachinePrecision] + N[(N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.6e-25

   1. Initial program 59.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 60.1

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

   8. Applied rewrites 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 72.9

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

   10. Applied rewrites 72.9%

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

   if 2.6e-25 < angle < 6.00000000000000037e153

   1. Initial program 45.7%

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lift-PI.f64 43.2

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   10. Applied rewrites 43.2%

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   if 6.00000000000000037e153 < angle

   1. Initial program 32.1%

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 45.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 39.0%

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(0.5 \cdot \pi\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 64.5% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(a + b\right)\\ t_1 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.5 \cdot 10^{+63}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot t\_0\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;angle\_m \leq 2.85 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot \cos t\_1\right) \cdot \sin t\_1}{b}, \frac{-2}{b}, \sin \left(2 \cdot t\_1\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(t\_0 \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (+ a b))) (t_1 (* (* PI angle_m) 0.005555555555555556)))
   (*
    angle_s
    (if (<= angle_m 2.5e+63)
      (* (* (* 0.011111111111111112 angle_m) t_0) (- b a))
      (if (<= angle_m 2.85e+259)
        (*
         (fma
          (/ (* (* (* a a) (cos t_1)) (sin t_1)) b)
          (/ -2.0 b)
          (sin (* 2.0 t_1)))
         (* b b))
        (* (* 0.011111111111111112 angle_m) (* t_0 (- b a))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (a + b);
	double t_1 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double tmp;
	if (angle_m <= 2.5e+63) {
		tmp = ((0.011111111111111112 * angle_m) * t_0) * (b - a);
	} else if (angle_m <= 2.85e+259) {
		tmp = fma(((((a * a) * cos(t_1)) * sin(t_1)) / b), (-2.0 / b), sin((2.0 * t_1))) * (b * b);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (t_0 * (b - a));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(a + b))
	t_1 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	tmp = 0.0
	if (angle_m <= 2.5e+63)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * t_0) * Float64(b - a));
	elseif (angle_m <= 2.85e+259)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(a * a) * cos(t_1)) * sin(t_1)) / b), Float64(-2.0 / b), sin(Float64(2.0 * t_1))) * Float64(b * b));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(t_0 * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.5e+63], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.85e+259], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * N[(-2.0 / b), $MachinePrecision] + N[Sin[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(t$95$0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.50000000000000005e63

   1. Initial program 59.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 59.9

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

   8. Applied rewrites 59.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 72.0

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

   10. Applied rewrites 72.0%

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

   if 2.50000000000000005e63 < angle < 2.85e259

   1. Initial program 37.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

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

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-*.f64 37.3

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

   7. Applied rewrites 37.3%

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

   if 2.85e259 < angle

   1. Initial program 20.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 42.8

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

   8. Applied rewrites 42.8%

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

Alternative 6: 63.6% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ t_1 := \sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right)\\ t_2 := \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(t\_2 \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot t\_1\right) \cdot t\_2\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(t\_1 \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right) \cdot \left(b - a\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556))
        (t_1 (sin (fma 0.5 PI t_0)))
        (t_2 (sin t_0)))
   (*
    angle_s
    (if (<= a 1.2e+148)
      (fma
       (* 2.0 (fma (* t_2 b) (sin (* 0.5 PI)) (* (* (* 0.0 a) t_1) t_2)))
       b
       (* (* -2.0 (* a a)) (* t_1 t_2)))
      (* (* (* 0.011111111111111112 angle_m) (* PI (+ a b))) (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double t_1 = sin(fma(0.5, ((double) M_PI), t_0));
	double t_2 = sin(t_0);
	double tmp;
	if (a <= 1.2e+148) {
		tmp = fma((2.0 * fma((t_2 * b), sin((0.5 * ((double) M_PI))), (((0.0 * a) * t_1) * t_2))), b, ((-2.0 * (a * a)) * (t_1 * t_2)));
	} else {
		tmp = ((0.011111111111111112 * angle_m) * (((double) M_PI) * (a + b))) * (b - a);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	t_1 = sin(fma(0.5, pi, t_0))
	t_2 = sin(t_0)
	tmp = 0.0
	if (a <= 1.2e+148)
		tmp = fma(Float64(2.0 * fma(Float64(t_2 * b), sin(Float64(0.5 * pi)), Float64(Float64(Float64(0.0 * a) * t_1) * t_2))), b, Float64(Float64(-2.0 * Float64(a * a)) * Float64(t_1 * t_2)));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(pi * Float64(a + b))) * Float64(b - a));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 1.2e+148], N[(N[(2.0 * N[(N[(t$95$2 * b), $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(0.0 * a), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.19999999999999997e148

   1. Initial program 55.6%

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lift-PI.f64 62.1

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   10. Applied rewrites 62.1%

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b, \sin \left(0.5 \cdot \pi\right), \left(\left(0 \cdot a\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   if 1.19999999999999997e148 < a

   1. Initial program 44.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 60.0

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

   8. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 74.2

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

   10. Applied rewrites 74.2%

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

Alternative 7: 63.4% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot angle\_m, 0.5 \cdot \pi\right)\right)\\ t_1 := \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\\ t_2 := \pi \cdot \left(a + b\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.7 \cdot 10^{+68}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot t\_2\right) \cdot \left(b - a\right)\\ \mathbf{elif}\;angle\_m \leq 2.15 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(t\_1 \cdot \left(a \cdot a\right)\right) \cdot t\_0}{b \cdot b}, -2, \left(t\_1 \cdot 2\right) \cdot t\_0\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(t\_2 \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (fma -0.005555555555555556 (* PI angle_m) (* 0.5 PI))))
        (t_1 (sin (* (* PI angle_m) 0.005555555555555556)))
        (t_2 (* PI (+ a b))))
   (*
    angle_s
    (if (<= angle_m 4.7e+68)
      (* (* (* 0.011111111111111112 angle_m) t_2) (- b a))
      (if (<= angle_m 2.15e+154)
        (*
         (fma (/ (* (* t_1 (* a a)) t_0) (* b b)) -2.0 (* (* t_1 2.0) t_0))
         (* b b))
        (* (* 0.011111111111111112 angle_m) (* t_2 (- b a))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin(fma(-0.005555555555555556, (((double) M_PI) * angle_m), (0.5 * ((double) M_PI))));
	double t_1 = sin(((((double) M_PI) * angle_m) * 0.005555555555555556));
	double t_2 = ((double) M_PI) * (a + b);
	double tmp;
	if (angle_m <= 4.7e+68) {
		tmp = ((0.011111111111111112 * angle_m) * t_2) * (b - a);
	} else if (angle_m <= 2.15e+154) {
		tmp = fma((((t_1 * (a * a)) * t_0) / (b * b)), -2.0, ((t_1 * 2.0) * t_0)) * (b * b);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (t_2 * (b - a));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(fma(-0.005555555555555556, Float64(pi * angle_m), Float64(0.5 * pi)))
	t_1 = sin(Float64(Float64(pi * angle_m) * 0.005555555555555556))
	t_2 = Float64(pi * Float64(a + b))
	tmp = 0.0
	if (angle_m <= 4.7e+68)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * t_2) * Float64(b - a));
	elseif (angle_m <= 2.15e+154)
		tmp = Float64(fma(Float64(Float64(Float64(t_1 * Float64(a * a)) * t_0) / Float64(b * b)), -2.0, Float64(Float64(t_1 * 2.0) * t_0)) * Float64(b * b));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(t_2 * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 4.7e+68], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.15e+154], N[(N[(N[(N[(N[(t$95$1 * N[(a * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(t$95$1 * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(t$95$2 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 4.6999999999999996e68

   1. Initial program 59.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 60.0

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

   8. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 72.0

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

   10. Applied rewrites 72.0%

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

   if 4.6999999999999996e68 < angle < 2.1499999999999999e154

   1. Initial program 33.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 33.3%

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

   if 2.1499999999999999e154 < angle

   1. Initial program 29.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 8: 63.3% accurate, N/A× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(a + b\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot t\_0\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(t\_0 \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (+ a b))))
   (*
    angle_s
    (if (<= angle_m 4.2e-25)
      (* (* (* 0.011111111111111112 angle_m) t_0) (- b a))
      (* (* 0.011111111111111112 angle_m) (* t_0 (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (a + b);
	double tmp;
	if (angle_m <= 4.2e-25) {
		tmp = ((0.011111111111111112 * angle_m) * t_0) * (b - a);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (t_0 * (b - a));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (a + b);
	double tmp;
	if (angle_m <= 4.2e-25) {
		tmp = ((0.011111111111111112 * angle_m) * t_0) * (b - a);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (t_0 * (b - a));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (a + b)
	tmp = 0
	if angle_m <= 4.2e-25:
		tmp = ((0.011111111111111112 * angle_m) * t_0) * (b - a)
	else:
		tmp = (0.011111111111111112 * angle_m) * (t_0 * (b - a))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(a + b))
	tmp = 0.0
	if (angle_m <= 4.2e-25)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * t_0) * Float64(b - a));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(t_0 * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (a + b);
	tmp = 0.0;
	if (angle_m <= 4.2e-25)
		tmp = ((0.011111111111111112 * angle_m) * t_0) * (b - a);
	else
		tmp = (0.011111111111111112 * angle_m) * (t_0 * (b - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 4.2e-25], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(t$95$0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.20000000000000005e-25

   1. Initial program 59.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 73.1

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

   10. Applied rewrites 73.1%

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

   if 4.20000000000000005e-25 < angle

   1. Initial program 40.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   7. 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. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 35.4

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

   8. Applied rewrites 35.4%

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

Alternative 9: 54.1% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 50.1%

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

6. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
7. 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. pow2 N/A

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

   3. pow2 N/A

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

   4. difference-of-squares-rev N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lift--.f64 52.8

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

8. Applied rewrites 52.8%

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

Reproduce

herbie shell --seed 2025065 
(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)))))