ab-angle->ABCF B

Percentage Accurate: 55.5% → 65.5%

Time: 6.7s

Alternatives: 14

Speedup: 2.6×

• 32.8% 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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ t_1 := \left(\pi \cdot 0.005555555555555556\right) \cdot angle\\ t_2 := \sin t\_1 \cdot \cos t\_1\\ \mathbf{if}\;a\_m \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, t\_2, t\_2 \cdot \left(0 \cdot a\_m\right)\right), b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot t\_2\right)\\ \mathbf{elif}\;a\_m \leq 2.95 \cdot 10^{+231}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right) \cdot a\_m\right) \cdot a\_m\right) \cdot -2\right) \cdot \cos t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right)\right) \cdot \sin t\_0\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0)))
        (t_1 (* (* PI 0.005555555555555556) angle))
        (t_2 (* (sin t_1) (cos t_1))))
   (if (<= a_m 9.5e+153)
     (fma
      (* 2.0 (fma b t_2 (* t_2 (* 0.0 a_m))))
      b
      (* (* -2.0 (* a_m a_m)) t_2))
     (if (<= a_m 2.95e+231)
       (*
        (*
         (*
          (*
           (*
            (fma
             PI
             0.005555555555555556
             (* (* -2.8577960676726107e-8 (* angle angle)) (* (* PI PI) PI)))
            angle)
           a_m)
          a_m)
         -2.0)
        (cos t_0))
       (*
        (* (* 2.0 (* (- b a_m) (+ a_m b))) (sin t_0))
        (cos (* PI (* 0.005555555555555556 angle))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double t_1 = (((double) M_PI) * 0.005555555555555556) * angle;
	double t_2 = sin(t_1) * cos(t_1);
	double tmp;
	if (a_m <= 9.5e+153) {
		tmp = fma((2.0 * fma(b, t_2, (t_2 * (0.0 * a_m)))), b, ((-2.0 * (a_m * a_m)) * t_2));
	} else if (a_m <= 2.95e+231) {
		tmp = ((((fma(((double) M_PI), 0.005555555555555556, ((-2.8577960676726107e-8 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) * angle) * a_m) * a_m) * -2.0) * cos(t_0);
	} else {
		tmp = ((2.0 * ((b - a_m) * (a_m + b))) * sin(t_0)) * cos((((double) M_PI) * (0.005555555555555556 * angle)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	t_1 = Float64(Float64(pi * 0.005555555555555556) * angle)
	t_2 = Float64(sin(t_1) * cos(t_1))
	tmp = 0.0
	if (a_m <= 9.5e+153)
		tmp = fma(Float64(2.0 * fma(b, t_2, Float64(t_2 * Float64(0.0 * a_m)))), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * t_2));
	elseif (a_m <= 2.95e+231)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(pi, 0.005555555555555556, Float64(Float64(-2.8577960676726107e-8 * Float64(angle * angle)) * Float64(Float64(pi * pi) * pi))) * angle) * a_m) * a_m) * -2.0) * cos(t_0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(a_m + b))) * sin(t_0)) * cos(Float64(pi * Float64(0.005555555555555556 * angle))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t$95$1], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 9.5e+153], N[(N[(2.0 * N[(b * t$95$2 + N[(t$95$2 * N[(0.0 * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 2.95e+231], N[(N[(N[(N[(N[(N[(Pi * 0.005555555555555556 + N[(N[(-2.8577960676726107e-8 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 9.4999999999999995e153

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 62.4%

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

   if 9.4999999999999995e153 < a < 2.95e231

   1. Initial program 55.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 36.3%

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

   6. Applied rewrites 41.3%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 39.6%

      \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -2\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if 2.95e231 < a

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.5

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

   3. Applied rewrites 55.5%

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

   4. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 58.9

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

   6. Applied rewrites 58.9%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 63.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot 0.005555555555555556\right) \cdot angle\\ \mathbf{if}\;angle \leq 1.45 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \left(\left(angle \cdot angle\right) \cdot b\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right) \cdot angle\right), b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (* PI 0.005555555555555556) angle)))
   (if (<= angle 1.45e+104)
     (fma
      (*
       2.0
       (*
        (fma
         (* 0.005555555555555556 b)
         PI
         (* (* (* angle angle) b) (* (* (* PI PI) PI) -1.1431184270690443e-7)))
        angle))
      b
      (* (* -2.0 (* a_m a_m)) (* (sin t_0) (cos t_0))))
     (*
      (* (* 2.0 (* (- b a_m) (+ a_m b))) (sin (* PI (/ angle 180.0))))
      (cos (* PI (* 0.005555555555555556 angle)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (((double) M_PI) * 0.005555555555555556) * angle;
	double tmp;
	if (angle <= 1.45e+104) {
		tmp = fma((2.0 * (fma((0.005555555555555556 * b), ((double) M_PI), (((angle * angle) * b) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7))) * angle)), b, ((-2.0 * (a_m * a_m)) * (sin(t_0) * cos(t_0))));
	} else {
		tmp = ((2.0 * ((b - a_m) * (a_m + b))) * sin((((double) M_PI) * (angle / 180.0)))) * cos((((double) M_PI) * (0.005555555555555556 * angle)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(pi * 0.005555555555555556) * angle)
	tmp = 0.0
	if (angle <= 1.45e+104)
		tmp = fma(Float64(2.0 * Float64(fma(Float64(0.005555555555555556 * b), pi, Float64(Float64(Float64(angle * angle) * b) * Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7))) * angle)), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * Float64(sin(t_0) * cos(t_0))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(a_m + b))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(pi * Float64(0.005555555555555556 * angle))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]}, If[LessEqual[angle, 1.45e+104], N[(N[(2.0 * N[(N[(N[(0.005555555555555556 * b), $MachinePrecision] * Pi + N[(N[(N[(angle * angle), $MachinePrecision] * b), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.4499999999999999e104

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 62.4%

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

   11. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 55.5%

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

   if 1.4499999999999999e104 < angle

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.5

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

   3. Applied rewrites 55.5%

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

   4. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 58.9

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

   6. Applied rewrites 58.9%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot 0.005555555555555556\right) \cdot angle\\ \mathbf{if}\;angle \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot b\right) \cdot angle\right) \cdot 0.011111111111111112, b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (* PI 0.005555555555555556) angle)))
   (if (<= angle 1.2e-95)
     (fma
      (* (* (* PI b) angle) 0.011111111111111112)
      b
      (* (* -2.0 (* a_m a_m)) (* (sin t_0) (cos t_0))))
     (*
      (* (* 2.0 (* (- b a_m) (+ a_m b))) (sin (* PI (/ angle 180.0))))
      (cos (* PI (* 0.005555555555555556 angle)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (((double) M_PI) * 0.005555555555555556) * angle;
	double tmp;
	if (angle <= 1.2e-95) {
		tmp = fma((((((double) M_PI) * b) * angle) * 0.011111111111111112), b, ((-2.0 * (a_m * a_m)) * (sin(t_0) * cos(t_0))));
	} else {
		tmp = ((2.0 * ((b - a_m) * (a_m + b))) * sin((((double) M_PI) * (angle / 180.0)))) * cos((((double) M_PI) * (0.005555555555555556 * angle)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(pi * 0.005555555555555556) * angle)
	tmp = 0.0
	if (angle <= 1.2e-95)
		tmp = fma(Float64(Float64(Float64(pi * b) * angle) * 0.011111111111111112), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * Float64(sin(t_0) * cos(t_0))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(a_m + b))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(pi * Float64(0.005555555555555556 * angle))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]}, If[LessEqual[angle, 1.2e-95], N[(N[(N[(N[(Pi * b), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.2e-95

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 62.4%

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

   11. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 58.3

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

   13. Applied rewrites 58.3%

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

   if 1.2e-95 < angle

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.5

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

   3. Applied rewrites 55.5%

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

   4. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 58.9

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

   6. Applied rewrites 58.9%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 59.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ \mathbf{if}\;angle \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, t\_0, t\_0 \cdot \left(0 \cdot a\_m\right)\right), b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (* angle PI) 0.005555555555555556)))
   (if (<= angle 1.2e-95)
     (fma
      (* 2.0 (fma b t_0 (* t_0 (* 0.0 a_m))))
      b
      (* (* -2.0 (* a_m a_m)) t_0))
     (*
      (* (* 2.0 (* (- b a_m) (+ a_m b))) (sin (* PI (/ angle 180.0))))
      (cos (* PI (* 0.005555555555555556 angle)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if (angle <= 1.2e-95) {
		tmp = fma((2.0 * fma(b, t_0, (t_0 * (0.0 * a_m)))), b, ((-2.0 * (a_m * a_m)) * t_0));
	} else {
		tmp = ((2.0 * ((b - a_m) * (a_m + b))) * sin((((double) M_PI) * (angle / 180.0)))) * cos((((double) M_PI) * (0.005555555555555556 * angle)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(angle * pi) * 0.005555555555555556)
	tmp = 0.0
	if (angle <= 1.2e-95)
		tmp = fma(Float64(2.0 * fma(b, t_0, Float64(t_0 * Float64(0.0 * a_m)))), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * t_0));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a_m) * Float64(a_m + b))) * sin(Float64(pi * Float64(angle / 180.0)))) * cos(Float64(pi * Float64(0.005555555555555556 * angle))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[angle, 1.2e-95], N[(N[(2.0 * N[(b * t$95$0 + N[(t$95$0 * N[(0.0 * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.2e-95

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 62.4%

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

   11. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 58.3

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 58.3

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

   13. Applied rewrites 58.3%

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

   14. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 58.2

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 58.2

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

   16. Applied rewrites 58.2%

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

   17. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 55.2

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 55.2

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

   19. Applied rewrites 55.2%

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

   if 1.2e-95 < angle

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 55.5

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

   3. Applied rewrites 55.5%

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

   4. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 58.9

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

   6. Applied rewrites 58.9%

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 59.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ t_1 := \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 2\\ t_2 := t\_1 \cdot \left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right)\\ \mathbf{if}\;angle \leq 3.1 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, t\_0, t\_0 \cdot \left(0 \cdot a\_m\right)\right), b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot t\_0\right)\\ \mathbf{elif}\;angle \leq 4.8 \cdot 10^{+104}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;angle \leq 1.5 \cdot 10^{+187}:\\ \;\;\;\;t\_1 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (* angle PI) 0.005555555555555556))
        (t_1 (* (* (+ b a_m) (- b a_m)) 2.0))
        (t_2
         (*
          t_1
          (*
           (fma
            PI
            0.005555555555555556
            (* (* (* (* PI PI) PI) -1.1431184270690443e-7) (* angle angle)))
           angle))))
   (if (<= angle 3.1e+96)
     (fma
      (* 2.0 (fma b t_0 (* t_0 (* 0.0 a_m))))
      b
      (* (* -2.0 (* a_m a_m)) t_0))
     (if (<= angle 4.8e+104)
       t_2
       (if (<= angle 1.5e+187)
         (* t_1 (* (* PI 0.005555555555555556) angle))
         t_2)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double t_1 = ((b + a_m) * (b - a_m)) * 2.0;
	double t_2 = t_1 * (fma(((double) M_PI), 0.005555555555555556, ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7) * (angle * angle))) * angle);
	double tmp;
	if (angle <= 3.1e+96) {
		tmp = fma((2.0 * fma(b, t_0, (t_0 * (0.0 * a_m)))), b, ((-2.0 * (a_m * a_m)) * t_0));
	} else if (angle <= 4.8e+104) {
		tmp = t_2;
	} else if (angle <= 1.5e+187) {
		tmp = t_1 * ((((double) M_PI) * 0.005555555555555556) * angle);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(angle * pi) * 0.005555555555555556)
	t_1 = Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * 2.0)
	t_2 = Float64(t_1 * Float64(fma(pi, 0.005555555555555556, Float64(Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7) * Float64(angle * angle))) * angle))
	tmp = 0.0
	if (angle <= 3.1e+96)
		tmp = fma(Float64(2.0 * fma(b, t_0, Float64(t_0 * Float64(0.0 * a_m)))), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * t_0));
	elseif (angle <= 4.8e+104)
		tmp = t_2;
	elseif (angle <= 1.5e+187)
		tmp = Float64(t_1 * Float64(Float64(pi * 0.005555555555555556) * angle));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(Pi * 0.005555555555555556 + N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision] * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 3.1e+96], N[(N[(2.0 * N[(b * t$95$0 + N[(t$95$0 * N[(0.0 * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 4.8e+104], t$95$2, If[LessEqual[angle, 1.5e+187], N[(t$95$1 * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0999999999999998e96

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 62.4%

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

   11. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 58.3

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 58.3

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

   13. Applied rewrites 58.3%

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

   14. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 58.2

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 58.2

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

   16. Applied rewrites 58.2%

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

   17. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 55.2

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 55.2

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

   19. Applied rewrites 55.2%

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

   if 3.0999999999999998e96 < angle < 4.8e104 or 1.5e187 < angle

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 53.6%

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

   if 4.8e104 < angle < 1.5e187

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 55.5

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

   10. Applied rewrites 55.5%

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

Alternative 6: 59.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(angle \cdot \pi\right) \cdot 0.005555555555555556\\ \mathbf{if}\;angle \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, t\_0, t\_0 \cdot \left(0 \cdot a\_m\right)\right), b, \left(-2 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot \sin \left(2 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (* angle PI) 0.005555555555555556)))
   (if (<= angle 1.2e-95)
     (fma
      (* 2.0 (fma b t_0 (* t_0 (* 0.0 a_m))))
      b
      (* (* -2.0 (* a_m a_m)) t_0))
     (*
      (* (- b a_m) (+ a_m b))
      (sin (* 2.0 (* (* PI 0.005555555555555556) angle)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = (angle * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if (angle <= 1.2e-95) {
		tmp = fma((2.0 * fma(b, t_0, (t_0 * (0.0 * a_m)))), b, ((-2.0 * (a_m * a_m)) * t_0));
	} else {
		tmp = ((b - a_m) * (a_m + b)) * sin((2.0 * ((((double) M_PI) * 0.005555555555555556) * angle)));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(angle * pi) * 0.005555555555555556)
	tmp = 0.0
	if (angle <= 1.2e-95)
		tmp = fma(Float64(2.0 * fma(b, t_0, Float64(t_0 * Float64(0.0 * a_m)))), b, Float64(Float64(-2.0 * Float64(a_m * a_m)) * t_0));
	else
		tmp = Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * sin(Float64(2.0 * Float64(Float64(pi * 0.005555555555555556) * angle))));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[angle, 1.2e-95], N[(N[(2.0 * N[(b * t$95$0 + N[(t$95$0 * N[(0.0 * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(-2.0 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.2e-95

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 62.4%

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

   11. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 58.3

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 58.3

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

   13. Applied rewrites 58.3%

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

   14. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 58.2

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 58.2

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

   16. Applied rewrites 58.2%

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

   17. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 55.2

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 55.2

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

   19. Applied rewrites 55.2%

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

   if 1.2e-95 < angle

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   9. Applied rewrites 59.2%

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

Alternative 7: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot b\right) \cdot b\right) \cdot 2\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -5e-232)
   (* (* -0.011111111111111112 a_m) (* a_m (* angle PI)))
   (* (* (* (* (sin (* (* 0.005555555555555556 PI) angle)) b) b) 2.0) 1.0)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -5e-232) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * ((double) M_PI)));
	} else {
		tmp = (((sin(((0.005555555555555556 * ((double) M_PI)) * angle)) * b) * b) * 2.0) * 1.0;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -5e-232) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * Math.PI));
	} else {
		tmp = (((Math.sin(((0.005555555555555556 * Math.PI) * angle)) * b) * b) * 2.0) * 1.0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -5e-232:
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * math.pi))
	else:
		tmp = (((math.sin(((0.005555555555555556 * math.pi) * angle)) * b) * b) * 2.0) * 1.0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -5e-232)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(a_m * Float64(angle * pi)));
	else
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * pi) * angle)) * b) * b) * 2.0) * 1.0);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -5e-232)
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * pi));
	else
		tmp = (((sin(((0.005555555555555556 * pi) * angle)) * b) * b) * 2.0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-232], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.9999999999999999e-232

   1. Initial program 55.5%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 35.3

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

   7. Applied rewrites 35.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.4

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

   9. Applied rewrites 35.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 38.6

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

       8. lift-PI.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 38.6

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

   11. Applied rewrites 38.6%

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

   if -4.9999999999999999e-232 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in a around 0

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

       1. associate-*l* N/A

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

       2. metadata-eval N/A

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

       3. mult-flip N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 36.0%

       \[\leadsto \color{blue}{\left(\left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right)} \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 40.7%

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

Alternative 8: 58.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 2\\ t_1 := t\_0 \cdot \left(\mathsf{fma}\left(\pi, 0.005555555555555556, \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot angle\right)\\ \mathbf{if}\;angle \leq 4.8 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;angle \leq 1.5 \cdot 10^{+187}:\\ \;\;\;\;t\_0 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (let* ((t_0 (* (* (+ b a_m) (- b a_m)) 2.0))
        (t_1
         (*
          t_0
          (*
           (fma
            PI
            0.005555555555555556
            (* (* (* (* PI PI) PI) -1.1431184270690443e-7) (* angle angle)))
           angle))))
   (if (<= angle 4.8e+104)
     t_1
     (if (<= angle 1.5e+187)
       (* t_0 (* (* PI 0.005555555555555556) angle))
       t_1))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double t_0 = ((b + a_m) * (b - a_m)) * 2.0;
	double t_1 = t_0 * (fma(((double) M_PI), 0.005555555555555556, ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7) * (angle * angle))) * angle);
	double tmp;
	if (angle <= 4.8e+104) {
		tmp = t_1;
	} else if (angle <= 1.5e+187) {
		tmp = t_0 * ((((double) M_PI) * 0.005555555555555556) * angle);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle)
	t_0 = Float64(Float64(Float64(b + a_m) * Float64(b - a_m)) * 2.0)
	t_1 = Float64(t_0 * Float64(fma(pi, 0.005555555555555556, Float64(Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7) * Float64(angle * angle))) * angle))
	tmp = 0.0
	if (angle <= 4.8e+104)
		tmp = t_1;
	elseif (angle <= 1.5e+187)
		tmp = Float64(t_0 * Float64(Float64(pi * 0.005555555555555556) * angle));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(Pi * 0.005555555555555556 + N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision] * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 4.8e+104], t$95$1, If[LessEqual[angle, 1.5e+187], N[(t$95$0 * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.8e104 or 1.5e187 < angle

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 53.6%

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

   if 4.8e104 < angle < 1.5e187

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 55.5

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

   10. Applied rewrites 55.5%

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

Alternative 9: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -4 \cdot 10^{+196}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot \left(a\_m + b\right)\right) \cdot \left(b - a\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -4e+196)
   (* (* -0.011111111111111112 a_m) (* a_m (* angle PI)))
   (* (* (* (* PI (+ a_m b)) (- b a_m)) angle) 0.011111111111111112)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -4e+196) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * ((double) M_PI)));
	} else {
		tmp = (((((double) M_PI) * (a_m + b)) * (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 ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -4e+196) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * Math.PI));
	} else {
		tmp = (((Math.PI * (a_m + b)) * (b - a_m)) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -4e+196:
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * math.pi))
	else:
		tmp = (((math.pi * (a_m + b)) * (b - a_m)) * angle) * 0.011111111111111112
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -4e+196)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(a_m * Float64(angle * pi)));
	else
		tmp = Float64(Float64(Float64(Float64(pi * Float64(a_m + b)) * Float64(b - a_m)) * angle) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -4e+196)
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * pi));
	else
		tmp = (((pi * (a_m + b)) * (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[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+196], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -3.9999999999999998e196

   1. Initial program 55.5%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 35.3

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

   7. Applied rewrites 35.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.4

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

   9. Applied rewrites 35.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 38.6

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

       8. lift-PI.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 38.6

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

   11. Applied rewrites 38.6%

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

   if -3.9999999999999998e196 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.2%

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

   8. 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. Applied rewrites 55.4%

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

Alternative 10: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -1e+112)
   (* (* -0.011111111111111112 a_m) (* a_m (* angle PI)))
   (* (* (* angle PI) 0.011111111111111112) (* (+ b a_m) (- b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -1e+112) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * ((double) M_PI)));
	} else {
		tmp = ((angle * ((double) M_PI)) * 0.011111111111111112) * ((b + a_m) * (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 ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -1e+112) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * Math.PI));
	} else {
		tmp = ((angle * Math.PI) * 0.011111111111111112) * ((b + a_m) * (b - a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -1e+112:
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * math.pi))
	else:
		tmp = ((angle * math.pi) * 0.011111111111111112) * ((b + a_m) * (b - a_m))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -1e+112)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(a_m * Float64(angle * pi)));
	else
		tmp = Float64(Float64(Float64(angle * pi) * 0.011111111111111112) * Float64(Float64(b + a_m) * 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 ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -1e+112)
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * pi));
	else
		tmp = ((angle * pi) * 0.011111111111111112) * ((b + a_m) * (b - a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+112], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -9.9999999999999993e111

   1. Initial program 55.5%

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

   2. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 35.3

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

   7. Applied rewrites 35.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.4

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

   9. Applied rewrites 35.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 38.6

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

       8. lift-PI.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 38.6

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

   11. Applied rewrites 38.6%

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

   if -9.9999999999999993e111 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 55.5

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

   3. Applied rewrites 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right) \]

      5. mult-flip N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{180}\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)} \cdot \frac{1}{180}\right) \]

      13. lift-PI.f64 55.6

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\left(\color{blue}{\pi} \cdot angle\right) \cdot 0.005555555555555556\right) \]

   5. Applied rewrites 55.6%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \]

   7. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]

   10. Applied rewrites 62.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right), \left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right)\right) \]

       2. pow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

       5. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right) \]

       6. pow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right) \]

       7. pow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

   13. Applied rewrites 55.5%

       \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -1.6 \cdot 10^{+24}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -1.6e+24)
   (* (* -0.011111111111111112 a_m) (* a_m (* angle PI)))
   (* (* (* 0.011111111111111112 angle) PI) (* (+ b a_m) (- b a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -1.6e+24) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * ((double) M_PI)));
	} else {
		tmp = ((0.011111111111111112 * angle) * ((double) M_PI)) * ((b + a_m) * (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 ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -1.6e+24) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * Math.PI));
	} else {
		tmp = ((0.011111111111111112 * angle) * Math.PI) * ((b + a_m) * (b - a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -1.6e+24:
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * math.pi))
	else:
		tmp = ((0.011111111111111112 * angle) * math.pi) * ((b + a_m) * (b - a_m))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -1.6e+24)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(a_m * Float64(angle * pi)));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle) * pi) * Float64(Float64(b + a_m) * 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 ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -1.6e+24)
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * pi));
	else
		tmp = ((0.011111111111111112 * angle) * pi) * ((b + a_m) * (b - a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.6e+24], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -1.5999999999999999e24

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 52.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 35.3

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 35.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 35.4

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 35.4%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\pi} \cdot angle\right)\right) \]

       7. lower-*.f64 38.6

          \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot \color{blue}{angle}\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       12. lift-PI.f64 38.6

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) \]

   11. Applied rewrites 38.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \pi\right)}\right) \]

   if -1.5999999999999999e24 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 52.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\pi} \cdot \left(b \cdot b - a \cdot a\right)\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a} \cdot a\right) \]

      9. pow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - \color{blue}{a} \cdot a\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - a \cdot \color{blue}{a}\right) \]

      11. pow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)} \]

      13. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left({\color{blue}{b}}^{2} - {a}^{2}\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left({b}^{\color{blue}{2}} - {a}^{2}\right) \]

      17. pow2 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right) \]

      18. pow2 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right) \]

      19. difference-of-squares N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

      21. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right) \]

      22. lower--.f64 55.5

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right) \]

   6. Applied rewrites 55.5%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -5e-232)
   (* (* -0.011111111111111112 a_m) (* a_m (* angle PI)))
   (* (* 0.011111111111111112 angle) (* PI (* b b)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -5e-232) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * ((double) M_PI)));
	} else {
		tmp = (0.011111111111111112 * angle) * (((double) M_PI) * (b * b));
	}
	return tmp;
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -5e-232) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * Math.PI));
	} else {
		tmp = (0.011111111111111112 * angle) * (Math.PI * (b * b));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -5e-232:
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * math.pi))
	else:
		tmp = (0.011111111111111112 * angle) * (math.pi * (b * b))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -5e-232)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(a_m * Float64(angle * pi)));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(pi * Float64(b * b)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -5e-232)
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * pi));
	else
		tmp = (0.011111111111111112 * angle) * (pi * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-232], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.9999999999999999e-232

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 52.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 35.3

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 35.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 35.4

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 35.4%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\pi} \cdot angle\right)\right) \]

       7. lower-*.f64 38.6

          \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot \color{blue}{angle}\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       12. lift-PI.f64 38.6

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) \]

   11. Applied rewrites 38.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \pi\right)}\right) \]

   if -4.9999999999999999e-232 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 52.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left({b}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {b}^{\color{blue}{2}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {b}^{\color{blue}{2}}\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot {b}^{2}\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

      5. lift-*.f64 36.2

         \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

   7. Applied rewrites 36.2%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq -5 \cdot 10^{-232}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) -5e-232)
   (* (* -0.011111111111111112 a_m) (* a_m (* angle PI)))
   (* (* (* PI (* b b)) angle) 0.011111111111111112)))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= -5e-232) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * ((double) M_PI)));
	} else {
		tmp = ((((double) M_PI) * (b * b)) * 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 ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= -5e-232) {
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * Math.PI));
	} else {
		tmp = ((Math.PI * (b * b)) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= -5e-232:
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * math.pi))
	else:
		tmp = ((math.pi * (b * b)) * angle) * 0.011111111111111112
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= -5e-232)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(a_m * Float64(angle * pi)));
	else
		tmp = Float64(Float64(Float64(pi * Float64(b * b)) * angle) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= -5e-232)
		tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * pi));
	else
		tmp = ((pi * (b * b)) * angle) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-232], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4.9999999999999999e-232

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 52.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 35.3

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 35.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

      5. lower-*.f64 35.4

         \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   9. Applied rewrites 35.4%

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\pi} \cdot angle\right)\right) \]

       7. lower-*.f64 38.6

          \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot \color{blue}{angle}\right)\right) \]

       8. lift-PI.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

       12. lift-PI.f64 38.6

           \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) \]

   11. Applied rewrites 38.6%

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \pi\right)}\right) \]

   if -4.9999999999999999e-232 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      10. lower-*.f64 52.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      8. pow2 N/A

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      9. lift-*.f64 36.2

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 36.2%

      \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \color{blue}{0.011111111111111112} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 38.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle)
 :precision binary64
 (* (* -0.011111111111111112 a_m) (* a_m (* angle PI))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle) {
	return (-0.011111111111111112 * a_m) * (a_m * (angle * ((double) M_PI)));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle) {
	return (-0.011111111111111112 * a_m) * (a_m * (angle * Math.PI));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle):
	return (-0.011111111111111112 * a_m) * (a_m * (angle * math.pi))

Julia

a_m = abs(a)
function code(a_m, b, angle)
	return Float64(Float64(-0.011111111111111112 * a_m) * Float64(a_m * Float64(angle * pi)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle)
	tmp = (-0.011111111111111112 * a_m) * (a_m * (angle * pi));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_] := N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   10. lower-*.f64 52.1

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

4. Applied rewrites 52.1%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)} \]

5. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   4. pow2 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   8. lift-PI.f64 35.3

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

7. Applied rewrites 35.3%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

   5. lower-*.f64 35.4

      \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

9. Applied rewrites 35.4%

   \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot \color{blue}{angle}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot a\right) \cdot a\right) \cdot \left(\pi \cdot angle\right) \]

    4. associate-*l* N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \]

    6. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\pi} \cdot angle\right)\right) \]

    7. lower-*.f64 38.6

       \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(\pi \cdot \color{blue}{angle}\right)\right) \]

    8. lift-PI.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\frac{-1}{90} \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

    12. lift-PI.f64 38.6

        \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \left(angle \cdot \pi\right)\right) \]

11. Applied rewrites 38.6%

    \[\leadsto \left(-0.011111111111111112 \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(angle \cdot \pi\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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)))))