ab-angle->ABCF B

Percentage Accurate: 53.7% → 58.9%

Time: 12.7s

Alternatives: 11

Speedup: 3.4×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 58.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a\_m \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \frac{b}{a\_m \cdot a\_m}, -1\right)\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{elif}\;a\_m \leq 3 \cdot 10^{+236}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= a_m 3.3e-91)
     (* t_0 (* 2.0 (* b b)))
     (if (<= a_m 5.2e+153)
       (*
        (* (* t_0 (* a_m a_m)) (* 2.0 (fma b (/ b (* a_m a_m)) -1.0)))
        (cos (* angle (* PI 0.005555555555555556))))
       (if (<= a_m 3e+236)
         (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
         (*
          angle
          (*
           (* (+ b a_m) (- b a_m))
           (fma
            -1.7146776406035666e-7
            (* angle (* angle (* PI (* PI PI))))
            (* PI 0.011111111111111112)))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 3.3e-91) {
		tmp = t_0 * (2.0 * (b * b));
	} else if (a_m <= 5.2e+153) {
		tmp = ((t_0 * (a_m * a_m)) * (2.0 * fma(b, (b / (a_m * a_m)), -1.0))) * cos((angle * (((double) M_PI) * 0.005555555555555556)));
	} else if (a_m <= 3e+236) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = angle * (((b + a_m) * (b - a_m)) * fma(-1.7146776406035666e-7, (angle * (angle * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))), (((double) M_PI) * 0.011111111111111112)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 3.3e-91)
		tmp = Float64(t_0 * Float64(2.0 * Float64(b * b)));
	elseif (a_m <= 5.2e+153)
		tmp = Float64(Float64(Float64(t_0 * Float64(a_m * a_m)) * Float64(2.0 * fma(b, Float64(b / Float64(a_m * a_m)), -1.0))) * cos(Float64(angle * Float64(pi * 0.005555555555555556))));
	elseif (a_m <= 3e+236)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(angle * Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * fma(-1.7146776406035666e-7, Float64(angle * Float64(angle * Float64(pi * Float64(pi * pi)))), Float64(pi * 0.011111111111111112))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 3.3e-91], N[(t$95$0 * N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 5.2e+153], N[(N[(N[(t$95$0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(b * N[(b / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 3e+236], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.7146776406035666e-7 * N[(angle * N[(angle * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 3.30000000000000011e-91

   1. Initial program 55.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 42.0

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

   5. Simplified 42.0%

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

   6. Taylor expanded in angle around 0

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

   8. Simplified 43.6%

      \[\leadsto \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right) \cdot \color{blue}{1} \]

   if 3.30000000000000011e-91 < a < 5.1999999999999998e153

   1. Initial program 54.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   8. Simplified 56.6%

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

   9. Taylor expanded in angle around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-cos.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-PI.f64 58.7

          \[\leadsto \left(\left(\left(a \cdot a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right) \cdot 2\right)\right) \cdot \cos \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.005555555555555556\right)\right) \]

   11. Simplified 58.7%

       \[\leadsto \left(\left(\left(a \cdot a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right) \cdot 2\right)\right) \cdot \color{blue}{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)} \]

   if 5.1999999999999998e153 < a < 2.9999999999999998e236

   1. Initial program 24.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 38.3

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 38.3%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 57.2%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 73.3

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 73.3%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if 2.9999999999999998e236 < a

   1. Initial program 70.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 70.6

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f64 88.2

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto angle \cdot \left(\frac{-1}{5832000} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto angle \cdot \left(\left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right) + \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right) \]

       5. distribute-rgt-out N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)} \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       9. lower--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{5832000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. Simplified 88.2%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 0.011111111111111112 \cdot \pi\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot a\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+236}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b}^{2} - {a\_m}^{2}\\ t_1 := \pi \cdot \frac{angle}{180}\\ t_2 := \cos t\_1\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+297}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\left(\left(2 \cdot t\_0\right) \cdot \sin t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\_m\right) \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a_m 2.0)))
        (t_1 (* PI (/ angle 180.0)))
        (t_2 (cos t_1)))
   (if (<= t_0 -4e+297)
     (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
     (if (<= t_0 5e+282)
       (* (* (* 2.0 t_0) (sin t_1)) t_2)
       (*
        t_2
        (* angle (* 0.011111111111111112 (* (+ b a_m) (* PI (- b a_m))))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = pow(b, 2.0) - pow(a_m, 2.0);
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double t_2 = cos(t_1);
	double tmp;
	if (t_0 <= -4e+297) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else if (t_0 <= 5e+282) {
		tmp = ((2.0 * t_0) * sin(t_1)) * t_2;
	} else {
		tmp = t_2 * (angle * (0.011111111111111112 * ((b + a_m) * (((double) M_PI) * (b - a_m)))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = Math.pow(b, 2.0) - Math.pow(a_m, 2.0);
	double t_1 = Math.PI * (angle / 180.0);
	double t_2 = Math.cos(t_1);
	double tmp;
	if (t_0 <= -4e+297) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * Math.PI)));
	} else if (t_0 <= 5e+282) {
		tmp = ((2.0 * t_0) * Math.sin(t_1)) * t_2;
	} else {
		tmp = t_2 * (angle * (0.011111111111111112 * ((b + a_m) * (Math.PI * (b - a_m)))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = math.pow(b, 2.0) - math.pow(a_m, 2.0)
	t_1 = math.pi * (angle / 180.0)
	t_2 = math.cos(t_1)
	tmp = 0
	if t_0 <= -4e+297:
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * math.pi)))
	elif t_0 <= 5e+282:
		tmp = ((2.0 * t_0) * math.sin(t_1)) * t_2
	else:
		tmp = t_2 * (angle * (0.011111111111111112 * ((b + a_m) * (math.pi * (b - a_m)))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64((b ^ 2.0) - (a_m ^ 2.0))
	t_1 = Float64(pi * Float64(angle / 180.0))
	t_2 = cos(t_1)
	tmp = 0.0
	if (t_0 <= -4e+297)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	elseif (t_0 <= 5e+282)
		tmp = Float64(Float64(Float64(2.0 * t_0) * sin(t_1)) * t_2);
	else
		tmp = Float64(t_2 * Float64(angle * Float64(0.011111111111111112 * Float64(Float64(b + a_m) * Float64(pi * Float64(b - a_m))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = (b ^ 2.0) - (a_m ^ 2.0);
	t_1 = pi * (angle / 180.0);
	t_2 = cos(t_1);
	tmp = 0.0;
	if (t_0 <= -4e+297)
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * pi)));
	elseif (t_0 <= 5e+282)
		tmp = ((2.0 * t_0) * sin(t_1)) * t_2;
	else
		tmp = t_2 * (angle * (0.011111111111111112 * ((b + a_m) * (pi * (b - a_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, If[LessEqual[t$95$0, -4e+297], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+282], N[(N[(N[(2.0 * t$95$0), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$2 * N[(angle * N[(0.011111111111111112 * N[(N[(b + a$95$m), $MachinePrecision] * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.0000000000000001e297

   1. Initial program 54.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 49.5

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 54.9%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 76.0

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 76.0%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if -4.0000000000000001e297 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.99999999999999978e282

   1. Initial program 58.7%

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

   if 4.99999999999999978e282 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   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 angle around 0

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{-1}{17496000}}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{{angle}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{-1}{17496000}\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   5. Simplified 48.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -5.7155921353452215 \cdot 10^{-8}\right), \pi \cdot 0.011111111111111112\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      8. lower-PI.f64 63.5

         \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Simplified 63.5%

      \[\leadsto \left(angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{+297}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := {b}^{2} - {a\_m}^{2}\\ t_1 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+297}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\left(\left(2 \cdot t\_0\right) \cdot \sin t\_1\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t\_1 \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\_m\right) \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (pow a_m 2.0))) (t_1 (* PI (/ angle 180.0))))
   (if (<= t_0 -4e+297)
     (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
     (if (<= t_0 5e+282)
       (*
        (* (* 2.0 t_0) (sin t_1))
        (cos (* 0.005555555555555556 (* angle PI))))
       (*
        (cos t_1)
        (* angle (* 0.011111111111111112 (* (+ b a_m) (* PI (- b a_m))))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = pow(b, 2.0) - pow(a_m, 2.0);
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if (t_0 <= -4e+297) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else if (t_0 <= 5e+282) {
		tmp = ((2.0 * t_0) * sin(t_1)) * cos((0.005555555555555556 * (angle * ((double) M_PI))));
	} else {
		tmp = cos(t_1) * (angle * (0.011111111111111112 * ((b + a_m) * (((double) M_PI) * (b - a_m)))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double t_0 = Math.pow(b, 2.0) - Math.pow(a_m, 2.0);
	double t_1 = Math.PI * (angle / 180.0);
	double tmp;
	if (t_0 <= -4e+297) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * Math.PI)));
	} else if (t_0 <= 5e+282) {
		tmp = ((2.0 * t_0) * Math.sin(t_1)) * Math.cos((0.005555555555555556 * (angle * Math.PI)));
	} else {
		tmp = Math.cos(t_1) * (angle * (0.011111111111111112 * ((b + a_m) * (Math.PI * (b - a_m)))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	t_0 = math.pow(b, 2.0) - math.pow(a_m, 2.0)
	t_1 = math.pi * (angle / 180.0)
	tmp = 0
	if t_0 <= -4e+297:
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * math.pi)))
	elif t_0 <= 5e+282:
		tmp = ((2.0 * t_0) * math.sin(t_1)) * math.cos((0.005555555555555556 * (angle * math.pi)))
	else:
		tmp = math.cos(t_1) * (angle * (0.011111111111111112 * ((b + a_m) * (math.pi * (b - a_m)))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64((b ^ 2.0) - (a_m ^ 2.0))
	t_1 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (t_0 <= -4e+297)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	elseif (t_0 <= 5e+282)
		tmp = Float64(Float64(Float64(2.0 * t_0) * sin(t_1)) * cos(Float64(0.005555555555555556 * Float64(angle * pi))));
	else
		tmp = Float64(cos(t_1) * Float64(angle * Float64(0.011111111111111112 * Float64(Float64(b + a_m) * Float64(pi * Float64(b - a_m))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	t_0 = (b ^ 2.0) - (a_m ^ 2.0);
	t_1 = pi * (angle / 180.0);
	tmp = 0.0;
	if (t_0 <= -4e+297)
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * pi)));
	elseif (t_0 <= 5e+282)
		tmp = ((2.0 * t_0) * sin(t_1)) * cos((0.005555555555555556 * (angle * pi)));
	else
		tmp = cos(t_1) * (angle * (0.011111111111111112 * ((b + a_m) * (pi * (b - a_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+297], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+282], N[(N[(N[(2.0 * t$95$0), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[t$95$1], $MachinePrecision] * N[(angle * N[(0.011111111111111112 * N[(N[(b + a$95$m), $MachinePrecision] * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.0000000000000001e297

   1. Initial program 54.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 49.5

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 54.9%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 76.0

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 76.0%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if -4.0000000000000001e297 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.99999999999999978e282

   1. Initial program 58.7%

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

   3. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 58.6

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

   5. Simplified 58.6%

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

   if 4.99999999999999978e282 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   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 angle around 0

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{-1}{17496000}}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{{angle}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{-1}{17496000}\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   5. Simplified 48.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -5.7155921353452215 \cdot 10^{-8}\right), \pi \cdot 0.011111111111111112\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      8. lower-PI.f64 63.5

         \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Simplified 63.5%

      \[\leadsto \left(angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{+297}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2.3 \cdot 10^{+238}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\_m\right) \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2.3e+238)
   (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
   (*
    (cos (* PI (/ angle 180.0)))
    (* angle (* 0.011111111111111112 (* (+ b a_m) (* PI (- b a_m))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((pow(b, 2.0) - pow(a_m, 2.0)) <= -2.3e+238) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = cos((((double) M_PI) * (angle / 180.0))) * (angle * (0.011111111111111112 * ((b + a_m) * (((double) M_PI) * (b - a_m)))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a_m, 2.0)) <= -2.3e+238) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * Math.PI)));
	} else {
		tmp = Math.cos((Math.PI * (angle / 180.0))) * (angle * (0.011111111111111112 * ((b + a_m) * (Math.PI * (b - a_m)))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a_m, 2.0)) <= -2.3e+238:
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * math.pi)))
	else:
		tmp = math.cos((math.pi * (angle / 180.0))) * (angle * (0.011111111111111112 * ((b + a_m) * (math.pi * (b - a_m)))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a_m ^ 2.0)) <= -2.3e+238)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(cos(Float64(pi * Float64(angle / 180.0))) * Float64(angle * Float64(0.011111111111111112 * Float64(Float64(b + a_m) * Float64(pi * Float64(b - a_m))))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a_m ^ 2.0)) <= -2.3e+238)
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * pi)));
	else
		tmp = cos((pi * (angle / 180.0))) * (angle * (0.011111111111111112 * ((b + a_m) * (pi * (b - a_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -2.3e+238], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(angle * N[(0.011111111111111112 * N[(N[(b + a$95$m), $MachinePrecision] * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -2.30000000000000003e238

   1. Initial program 54.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 49.0

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 49.0%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 53.8%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 72.0

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 72.0%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if -2.30000000000000003e238 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{-1}{17496000}}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{{angle}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{-1}{17496000}\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   5. Simplified 50.5%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -5.7155921353452215 \cdot 10^{-8}\right), \pi \cdot 0.011111111111111112\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      8. lower-PI.f64 56.9

         \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Simplified 56.9%

      \[\leadsto \left(angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2.3 \cdot 10^{+238}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\_m\right) \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2e-78)
   (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
   (* angle (* 0.011111111111111112 (* (+ b a_m) (* PI (- b a_m)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((pow(b, 2.0) - pow(a_m, 2.0)) <= -2e-78) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = angle * (0.011111111111111112 * ((b + a_m) * (((double) M_PI) * (b - a_m))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a_m, 2.0)) <= -2e-78) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * Math.PI)));
	} else {
		tmp = angle * (0.011111111111111112 * ((b + a_m) * (Math.PI * (b - a_m))));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a_m, 2.0)) <= -2e-78:
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * math.pi)))
	else:
		tmp = angle * (0.011111111111111112 * ((b + a_m) * (math.pi * (b - a_m))))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a_m ^ 2.0)) <= -2e-78)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(Float64(b + a_m) * Float64(pi * Float64(b - a_m)))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a_m ^ 2.0)) <= -2e-78)
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * pi)));
	else
		tmp = angle * (0.011111111111111112 * ((b + a_m) * (pi * (b - a_m))));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -2e-78], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(N[(b + a$95$m), $MachinePrecision] * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -2e-78

   1. Initial program 51.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 45.7

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 45.7%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 48.6%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 60.4

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 60.4%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if -2e-78 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 54.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \frac{-1}{17496000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \frac{-1}{17496000}}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + \color{blue}{{angle}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{-1}{17496000}\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) + {angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   5. Simplified 53.2%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -5.7155921353452215 \cdot 10^{-8}\right), \pi \cdot 0.011111111111111112\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      8. lower-PI.f64 61.0

         \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Simplified 61.0%

      \[\leadsto \left(angle \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   9. Taylor expanded in angle around 0

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

   11. Simplified 59.3%

       \[\leadsto \left(angle \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\_m\right) \cdot \left(\pi \cdot \left(b - a\_m\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -2e-78)
   (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
   (* (* (+ b a_m) (* PI (- b a_m))) (* angle 0.011111111111111112))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((pow(b, 2.0) - pow(a_m, 2.0)) <= -2e-78) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = ((b + a_m) * (((double) M_PI) * (b - a_m))) * (angle * 0.011111111111111112);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a_m, 2.0)) <= -2e-78) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * Math.PI)));
	} else {
		tmp = ((b + a_m) * (Math.PI * (b - a_m))) * (angle * 0.011111111111111112);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a_m, 2.0)) <= -2e-78:
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * math.pi)))
	else:
		tmp = ((b + a_m) * (math.pi * (b - a_m))) * (angle * 0.011111111111111112)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a_m ^ 2.0)) <= -2e-78)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(Float64(Float64(b + a_m) * Float64(pi * Float64(b - a_m))) * Float64(angle * 0.011111111111111112));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a_m ^ 2.0)) <= -2e-78)
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * pi)));
	else
		tmp = ((b + a_m) * (pi * (b - a_m))) * (angle * 0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -2e-78], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(Pi * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -2e-78

   1. Initial program 51.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 45.7

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 45.7%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 48.6%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 60.4

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 60.4%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if -2e-78 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 54.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 61.0

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 61.0%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f64 54.9

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 54.9%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

   9. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-*.f64 59.2

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

   11. Simplified 59.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-78}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) -5e-273)
   (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
   (* angle (* 0.011111111111111112 (* b (* b PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((pow(b, 2.0) - pow(a_m, 2.0)) <= -5e-273) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = angle * (0.011111111111111112 * (b * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a_m, 2.0)) <= -5e-273) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * Math.PI)));
	} else {
		tmp = angle * (0.011111111111111112 * (b * (b * Math.PI)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a_m, 2.0)) <= -5e-273:
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * math.pi)))
	else:
		tmp = angle * (0.011111111111111112 * (b * (b * math.pi)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a_m ^ 2.0)) <= -5e-273)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(b * Float64(b * pi))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a_m ^ 2.0)) <= -5e-273)
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * pi)));
	else
		tmp = angle * (0.011111111111111112 * (b * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -5e-273], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(b * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.99999999999999965e-273

   1. Initial program 53.2%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 47.6

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 47.6%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 50.2%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 60.6

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 60.6%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if -4.99999999999999965e-273 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 53.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 60.7

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 60.7%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f64 54.1

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 54.1%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

   9. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-*.f64 58.8

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

   11. Simplified 58.8%

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

   12. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 55.2

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

   14. Simplified 55.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-273}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a\_m}^{2} \leq 5 \cdot 10^{-235}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a\_m \cdot a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a_m 2.0)) 5e-235)
   (* -0.011111111111111112 (* PI (* angle (* a_m a_m))))
   (* angle (* 0.011111111111111112 (* b (* b PI))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((pow(b, 2.0) - pow(a_m, 2.0)) <= 5e-235) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a_m * a_m)));
	} else {
		tmp = angle * (0.011111111111111112 * (b * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a_m, 2.0)) <= 5e-235) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a_m * a_m)));
	} else {
		tmp = angle * (0.011111111111111112 * (b * (b * Math.PI)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a_m, 2.0)) <= 5e-235:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a_m * a_m)))
	else:
		tmp = angle * (0.011111111111111112 * (b * (b * math.pi)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a_m ^ 2.0)) <= 5e-235)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a_m * a_m))));
	else
		tmp = Float64(angle * Float64(0.011111111111111112 * Float64(b * Float64(b * pi))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a_m ^ 2.0)) <= 5e-235)
		tmp = -0.011111111111111112 * (pi * (angle * (a_m * a_m)));
	else
		tmp = angle * (0.011111111111111112 * (b * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 5e-235], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(0.011111111111111112 * N[(b * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.9999999999999998e-235

   1. Initial program 58.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 53.6

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 53.6%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f64 52.0

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 52.0%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

   9. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-*.f64 55.7

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

   11. Simplified 55.7%

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

   12. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 56.2

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

   14. Simplified 56.2%

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

   if 4.9999999999999998e-235 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 46.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 56.3

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f64 50.1

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 50.1%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

   9. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-*.f64 54.0

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

   11. Simplified 54.0%

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

   12. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 49.6

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

   14. Simplified 49.6%

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

Alternative 9: 58.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(t\_0 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \frac{b}{a\_m \cdot a\_m}, -1\right)\right)\\ \mathbf{elif}\;a\_m \leq 3 \cdot 10^{+236}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= a_m 3.3e-91)
     (* t_0 (* 2.0 (* b b)))
     (if (<= a_m 1.32e+154)
       (* (* t_0 (* a_m a_m)) (* 2.0 (fma b (/ b (* a_m a_m)) -1.0)))
       (if (<= a_m 3e+236)
         (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
         (*
          angle
          (*
           (* (+ b a_m) (- b a_m))
           (fma
            -1.7146776406035666e-7
            (* angle (* angle (* PI (* PI PI))))
            (* PI 0.011111111111111112)))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 3.3e-91) {
		tmp = t_0 * (2.0 * (b * b));
	} else if (a_m <= 1.32e+154) {
		tmp = (t_0 * (a_m * a_m)) * (2.0 * fma(b, (b / (a_m * a_m)), -1.0));
	} else if (a_m <= 3e+236) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = angle * (((b + a_m) * (b - a_m)) * fma(-1.7146776406035666e-7, (angle * (angle * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))), (((double) M_PI) * 0.011111111111111112)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (a_m <= 3.3e-91)
		tmp = Float64(t_0 * Float64(2.0 * Float64(b * b)));
	elseif (a_m <= 1.32e+154)
		tmp = Float64(Float64(t_0 * Float64(a_m * a_m)) * Float64(2.0 * fma(b, Float64(b / Float64(a_m * a_m)), -1.0)));
	elseif (a_m <= 3e+236)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(angle * Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * fma(-1.7146776406035666e-7, Float64(angle * Float64(angle * Float64(pi * Float64(pi * pi)))), Float64(pi * 0.011111111111111112))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 3.3e-91], N[(t$95$0 * N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.32e+154], N[(N[(t$95$0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(b * N[(b / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 3e+236], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.7146776406035666e-7 * N[(angle * N[(angle * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 3.30000000000000011e-91

   1. Initial program 55.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 42.0

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

   5. Simplified 42.0%

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

   6. Taylor expanded in angle around 0

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

   8. Simplified 43.6%

      \[\leadsto \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right) \cdot \color{blue}{1} \]

   if 3.30000000000000011e-91 < a < 1.31999999999999998e154

   1. Initial program 53.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.3

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

   5. Simplified 53.3%

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

   6. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   8. Simplified 55.6%

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

   9. Taylor expanded in angle around 0

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

   11. Simplified 54.0%

       \[\leadsto \left(\left(\left(a \cdot a\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right) \cdot 2\right)\right) \cdot \color{blue}{1} \]

   if 1.31999999999999998e154 < a < 2.9999999999999998e236

   1. Initial program 25.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 40.1

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 40.1%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 59.1%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 76.0

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 76.0%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if 2.9999999999999998e236 < a

   1. Initial program 70.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 70.6

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f64 88.2

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto angle \cdot \left(\frac{-1}{5832000} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto angle \cdot \left(\left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right) + \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right) \]

       5. distribute-rgt-out N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)} \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       9. lower--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{5832000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. Simplified 88.2%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 0.011111111111111112 \cdot \pi\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot a\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+236}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a\_m \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot \mathsf{fma}\left(b, \frac{b}{a\_m \cdot a\_m}, -1\right)\right) \cdot \left(0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right)\\ \mathbf{elif}\;a\_m \leq 3 \cdot 10^{+236}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= a_m 3.3e-91)
   (* (sin (* 0.005555555555555556 (* angle PI))) (* 2.0 (* b b)))
   (if (<= a_m 5.2e+153)
     (*
      (* (* angle PI) (fma b (/ b (* a_m a_m)) -1.0))
      (* 0.011111111111111112 (* a_m a_m)))
     (if (<= a_m 3e+236)
       (* a_m (* -0.011111111111111112 (* angle (* a_m PI))))
       (*
        angle
        (*
         (* (+ b a_m) (- b a_m))
         (fma
          -1.7146776406035666e-7
          (* angle (* angle (* PI (* PI PI))))
          (* PI 0.011111111111111112))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if (a_m <= 3.3e-91) {
		tmp = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (2.0 * (b * b));
	} else if (a_m <= 5.2e+153) {
		tmp = ((angle * ((double) M_PI)) * fma(b, (b / (a_m * a_m)), -1.0)) * (0.011111111111111112 * (a_m * a_m));
	} else if (a_m <= 3e+236) {
		tmp = a_m * (-0.011111111111111112 * (angle * (a_m * ((double) M_PI))));
	} else {
		tmp = angle * (((b + a_m) * (b - a_m)) * fma(-1.7146776406035666e-7, (angle * (angle * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))), (((double) M_PI) * 0.011111111111111112)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (a_m <= 3.3e-91)
		tmp = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(2.0 * Float64(b * b)));
	elseif (a_m <= 5.2e+153)
		tmp = Float64(Float64(Float64(angle * pi) * fma(b, Float64(b / Float64(a_m * a_m)), -1.0)) * Float64(0.011111111111111112 * Float64(a_m * a_m)));
	elseif (a_m <= 3e+236)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(angle * Float64(a_m * pi))));
	else
		tmp = Float64(angle * Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * fma(-1.7146776406035666e-7, Float64(angle * Float64(angle * Float64(pi * Float64(pi * pi)))), Float64(pi * 0.011111111111111112))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[a$95$m, 3.3e-91], N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 5.2e+153], N[(N[(N[(angle * Pi), $MachinePrecision] * N[(b * N[(b / N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 3e+236], N[(a$95$m * N[(-0.011111111111111112 * N[(angle * N[(a$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.7146776406035666e-7 * N[(angle * N[(angle * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < 3.30000000000000011e-91

   1. Initial program 55.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 42.0

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

   5. Simplified 42.0%

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

   6. Taylor expanded in angle around 0

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

   8. Simplified 43.6%

      \[\leadsto \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right) \cdot \color{blue}{1} \]

   if 3.30000000000000011e-91 < a < 5.1999999999999998e153

   1. Initial program 54.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{{b}^{2}}{{a}^{2}} - 1\right)\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{fma}\left(b, \frac{b}{{a}^{2}}, -1\right)}\right) \cdot \left(\frac{1}{90} \cdot {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(b, \color{blue}{\frac{b}{{a}^{2}}}, -1\right)\right) \cdot \left(\frac{1}{90} \cdot {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{\color{blue}{a \cdot a}}, -1\right)\right) \cdot \left(\frac{1}{90} \cdot {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{\color{blue}{a \cdot a}}, -1\right)\right) \cdot \left(\frac{1}{90} \cdot {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot {a}^{2}\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      17. unpow2 N/A

          \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      18. lower-*.f64 53.9

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right) \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right) \cdot \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   9. Taylor expanded in angle around 0

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

   11. Simplified 55.3%

       \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right) \cdot \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{1} \]

   if 5.1999999999999998e153 < a < 2.9999999999999998e236

   1. Initial program 24.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 38.3

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 38.3%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

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

   8. Simplified 57.2%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-PI.f64 73.3

           \[\leadsto \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot a\right)\right)\right)\right) \cdot 1 \]

   11. Simplified 73.3%

       \[\leadsto \color{blue}{\left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\right)} \cdot 1 \]

   if 2.9999999999999998e236 < a

   1. Initial program 70.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. difference-of-squares N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      15. lower--.f64 70.6

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f64 88.2

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 88.2%

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{angle \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto angle \cdot \left(\frac{-1}{5832000} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto angle \cdot \left(\left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right) + \color{blue}{\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right) \]

       5. distribute-rgt-out N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)} \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       9. lower--.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot \left(\frac{-1}{5832000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{5832000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. Simplified 88.2%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 0.011111111111111112 \cdot \pi\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot \mathsf{fma}\left(b, \frac{b}{a \cdot a}, -1\right)\right) \cdot \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+236}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{fma}\left(-1.7146776406035666 \cdot 10^{-7}, angle \cdot \left(angle \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 21.6× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* -0.011111111111111112 (* PI (* angle (* a_m a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return -0.011111111111111112 * (((double) M_PI) * (angle * (a_m * a_m)));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return -0.011111111111111112 * (Math.PI * (angle * (a_m * a_m)));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return -0.011111111111111112 * (math.pi * (angle * (a_m * a_m)))

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a_m * a_m))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = -0.011111111111111112 * (pi * (angle * (a_m * a_m)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. unpow2 N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   11. difference-of-squares N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   15. lower--.f64 54.8

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Simplified 54.8%

   \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Taylor expanded in angle around 0

   \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{-1}{64800} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{64800}} + 1\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left(\color{blue}{{angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right)} + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{angle \cdot angle}, \frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \color{blue}{\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

   11. lower-PI.f64 N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{90} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, \frac{-1}{64800} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

   12. lower-PI.f64 51.2

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

8. Simplified 51.2%

   \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(angle \cdot angle, -1.54320987654321 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right)} \]

9. Taylor expanded in angle around 0

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-+.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lower--.f64 N/A

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

    10. lower-PI.f64 N/A

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

    11. lower-*.f64 54.9

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

11. Simplified 54.9%

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

12. Taylor expanded in a around inf

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

    1. lower-*.f64 N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-PI.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 35.7

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

14. Simplified 35.7%

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

Reproduce

herbie shell --seed 2024215 
(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)))))