ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.9%

Time: 7.9s

Alternatives: 20

Speedup: 6.6×

• 32.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a\_m + b\right)\right) \cdot 2\right) \cdot \left(\left(b - a\_m\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot \sin \left(2 \cdot \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.55e+106)
    (*
     (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a_m b)) 2.0)
     (* (- b a_m) (cos (* -0.005555555555555556 (* angle_m PI)))))
    (*
     (* (- b a_m) (+ a_m b))
     (sin
      (*
       2.0
       (*
        (pow (cbrt PI) 2.0)
        (* (cbrt PI) (* 0.005555555555555556 angle_m)))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.55e+106) {
		tmp = ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a_m + b)) * 2.0) * ((b - a_m) * cos((-0.005555555555555556 * (angle_m * ((double) M_PI)))));
	} else {
		tmp = ((b - a_m) * (a_m + b)) * sin((2.0 * (pow(cbrt(((double) M_PI)), 2.0) * (cbrt(((double) M_PI)) * (0.005555555555555556 * angle_m)))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.55e+106) {
		tmp = ((Math.sin(((0.005555555555555556 * angle_m) * Math.PI)) * (a_m + b)) * 2.0) * ((b - a_m) * Math.cos((-0.005555555555555556 * (angle_m * Math.PI))));
	} else {
		tmp = ((b - a_m) * (a_m + b)) * Math.sin((2.0 * (Math.pow(Math.cbrt(Math.PI), 2.0) * (Math.cbrt(Math.PI) * (0.005555555555555556 * angle_m)))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.55e+106)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a_m + b)) * 2.0) * Float64(Float64(b - a_m) * cos(Float64(-0.005555555555555556 * Float64(angle_m * pi)))));
	else
		tmp = Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * sin(Float64(2.0 * Float64((cbrt(pi) ^ 2.0) * Float64(cbrt(pi) * Float64(0.005555555555555556 * angle_m))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.55e+106], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.55e106

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.6%

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

   if 1.55e106 < angle

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

   5. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right) \]

      12. pow1/3 N/A

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

      13. lift-PI.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-prod-up N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \frac{angle}{180}\right)\right)\right) \]

      20. lower-cbrt.f64 57.8

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \frac{angle}{180}\right)\right)\right) \]

      21. lift-/.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right) \]

      22. mult-flip N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \]

      24. *-commutative N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right) \]

      25. lift-*.f64 57.8

          \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Applied rewrites 57.8%

      \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{{\pi}^{\frac{2}{3}}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\color{blue}{\left(\frac{2}{3}\right)}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      3. pow-cbrt N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      4. lift-cbrt.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{2} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      5. lower-pow.f64 57.8

         \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   9. Applied rewrites 57.8%

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

Alternative 2: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-111}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(a\_m + b\right)\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{+201}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a\_m + b\right)\right) \cdot 2\right) \cdot \left(\left(b - a\_m\right) \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(\pi \cdot angle\_m\right) \cdot \left(b - a\_m\right)\right) \cdot \left(a\_m + b\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 2.9e-111)
    (* (* (sin (* 0.011111111111111112 (* angle_m PI))) (- b a_m)) (+ a_m b))
    (if (<= b 8.4e+201)
      (*
       (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a_m b)) 2.0)
       (* (- b a_m) 1.0))
      (* 0.011111111111111112 (* (* (* PI angle_m) (- b a_m)) (+ a_m b)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (b <= 2.9e-111) {
		tmp = (sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * (b - a_m)) * (a_m + b);
	} else if (b <= 8.4e+201) {
		tmp = ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a_m + b)) * 2.0) * ((b - a_m) * 1.0);
	} else {
		tmp = 0.011111111111111112 * (((((double) M_PI) * angle_m) * (b - a_m)) * (a_m + b));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (b <= 2.9e-111) {
		tmp = (Math.sin((0.011111111111111112 * (angle_m * Math.PI))) * (b - a_m)) * (a_m + b);
	} else if (b <= 8.4e+201) {
		tmp = ((Math.sin(((0.005555555555555556 * angle_m) * Math.PI)) * (a_m + b)) * 2.0) * ((b - a_m) * 1.0);
	} else {
		tmp = 0.011111111111111112 * (((Math.PI * angle_m) * (b - a_m)) * (a_m + b));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if b <= 2.9e-111:
		tmp = (math.sin((0.011111111111111112 * (angle_m * math.pi))) * (b - a_m)) * (a_m + b)
	elif b <= 8.4e+201:
		tmp = ((math.sin(((0.005555555555555556 * angle_m) * math.pi)) * (a_m + b)) * 2.0) * ((b - a_m) * 1.0)
	else:
		tmp = 0.011111111111111112 * (((math.pi * angle_m) * (b - a_m)) * (a_m + b))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (b <= 2.9e-111)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * Float64(b - a_m)) * Float64(a_m + b));
	elseif (b <= 8.4e+201)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a_m + b)) * 2.0) * Float64(Float64(b - a_m) * 1.0));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(pi * angle_m) * Float64(b - a_m)) * Float64(a_m + b)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (b <= 2.9e-111)
		tmp = (sin((0.011111111111111112 * (angle_m * pi))) * (b - a_m)) * (a_m + b);
	elseif (b <= 8.4e+201)
		tmp = ((sin(((0.005555555555555556 * angle_m) * pi)) * (a_m + b)) * 2.0) * ((b - a_m) * 1.0);
	else
		tmp = 0.011111111111111112 * (((pi * angle_m) * (b - a_m)) * (a_m + b));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 2.9e-111], N[(N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.4e+201], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < 2.90000000000000002e-111

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.9%

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

   if 2.90000000000000002e-111 < b < 8.3999999999999996e201

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 66.2%

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

   if 8.3999999999999996e201 < b

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

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

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

      11. lift-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 62.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 62.8

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 62.8

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

   6. Applied rewrites 62.8%

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

Alternative 3: 67.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(a\_m + b\right)\\ \mathbf{elif}\;angle\_m \leq 2.5 \cdot 10^{+211}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \mathsf{fma}\left(b, b, \left(-a\_m\right) \cdot a\_m\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \left(\pi + \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.6e+16)
    (* (* (sin (* 0.011111111111111112 (* angle_m PI))) (- b a_m)) (+ a_m b))
    (if (<= angle_m 2.5e+211)
      (* 0.011111111111111112 (* (* angle_m (fma b b (* (- a_m) a_m))) PI))
      (*
       (* (- b a_m) (+ a_m b))
       (sin (* (* 0.005555555555555556 angle_m) (+ PI PI))))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e+16) {
		tmp = (sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * (b - a_m)) * (a_m + b);
	} else if (angle_m <= 2.5e+211) {
		tmp = 0.011111111111111112 * ((angle_m * fma(b, b, (-a_m * a_m))) * ((double) M_PI));
	} else {
		tmp = ((b - a_m) * (a_m + b)) * sin(((0.005555555555555556 * angle_m) * (((double) M_PI) + ((double) M_PI))));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.6e+16)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * Float64(b - a_m)) * Float64(a_m + b));
	elseif (angle_m <= 2.5e+211)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * fma(b, b, Float64(Float64(-a_m) * a_m))) * pi));
	else
		tmp = Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * sin(Float64(Float64(0.005555555555555556 * angle_m) * Float64(pi + pi))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.6e+16], N[(N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.5e+211], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b * b + N[((-a$95$m) * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 4.6e16

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.9%

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

   if 4.6e16 < angle < 2.4999999999999998e211

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. lift-+.f64 N/A

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

      14. lift--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 54.9

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

   6. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      13. unpow2 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left({a}^{2}\right)\right)\right)\right) \cdot \pi\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left({a}^{2}\right)\right)\right) \cdot \pi\right) \]

      17. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left(a \cdot a\right)\right)\right) \cdot \pi\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      20. lower-neg.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 2.4999999999999998e211 < angle

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

   5. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. lower-+.f64 58.1

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

   7. Applied rewrites 58.1%

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

Alternative 4: 67.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 8.2 \cdot 10^{+204}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(\left(a\_m + b\right) \cdot \left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot 2\right)\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 8.2e+204)
    (*
     (*
      (- b a_m)
      (* (+ a_m b) (* (sin (* (* PI angle_m) 0.005555555555555556)) 2.0)))
     1.0)
    (*
     (* (- b a_m) (+ a_m b))
     (sin (* 2.0 (* (* 0.005555555555555556 PI) angle_m)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 8.2e+204) {
		tmp = ((b - a_m) * ((a_m + b) * (sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * 2.0))) * 1.0;
	} else {
		tmp = ((b - a_m) * (a_m + b)) * sin((2.0 * ((0.005555555555555556 * ((double) M_PI)) * angle_m)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 8.2e+204) {
		tmp = ((b - a_m) * ((a_m + b) * (Math.sin(((Math.PI * angle_m) * 0.005555555555555556)) * 2.0))) * 1.0;
	} else {
		tmp = ((b - a_m) * (a_m + b)) * Math.sin((2.0 * ((0.005555555555555556 * Math.PI) * angle_m)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 8.2e+204:
		tmp = ((b - a_m) * ((a_m + b) * (math.sin(((math.pi * angle_m) * 0.005555555555555556)) * 2.0))) * 1.0
	else:
		tmp = ((b - a_m) * (a_m + b)) * math.sin((2.0 * ((0.005555555555555556 * math.pi) * angle_m)))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 8.2e+204)
		tmp = Float64(Float64(Float64(b - a_m) * Float64(Float64(a_m + b) * Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * 2.0))) * 1.0);
	else
		tmp = Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * sin(Float64(2.0 * Float64(Float64(0.005555555555555556 * pi) * angle_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 8.2e+204)
		tmp = ((b - a_m) * ((a_m + b) * (sin(((pi * angle_m) * 0.005555555555555556)) * 2.0))) * 1.0;
	else
		tmp = ((b - a_m) * (a_m + b)) * sin((2.0 * ((0.005555555555555556 * pi) * angle_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 8.2e+204], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 8.19999999999999949e204

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 66.2%

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

   if 8.19999999999999949e204 < angle

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

   5. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 58.1

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

   7. Applied rewrites 58.1%

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

Alternative 5: 67.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(a\_m + b\right)\\ \mathbf{elif}\;angle\_m \leq 2.4 \cdot 10^{+211}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \mathsf{fma}\left(b, b, \left(-a\_m\right) \cdot a\_m\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.6e+16)
    (* (* (sin (* 0.011111111111111112 (* angle_m PI))) (- b a_m)) (+ a_m b))
    (if (<= angle_m 2.4e+211)
      (* 0.011111111111111112 (* (* angle_m (fma b b (* (- a_m) a_m))) PI))
      (*
       (* (- b a_m) (+ a_m b))
       (sin (* (* angle_m PI) 0.011111111111111112)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e+16) {
		tmp = (sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * (b - a_m)) * (a_m + b);
	} else if (angle_m <= 2.4e+211) {
		tmp = 0.011111111111111112 * ((angle_m * fma(b, b, (-a_m * a_m))) * ((double) M_PI));
	} else {
		tmp = ((b - a_m) * (a_m + b)) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.6e+16)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * Float64(b - a_m)) * Float64(a_m + b));
	elseif (angle_m <= 2.4e+211)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * fma(b, b, Float64(Float64(-a_m) * a_m))) * pi));
	else
		tmp = Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.6e+16], N[(N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.4e+211], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b * b + N[((-a$95$m) * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 4.6e16

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.9%

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

   if 4.6e16 < angle < 2.40000000000000018e211

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. lift-+.f64 N/A

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

      14. lift--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 54.9

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

   6. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      13. unpow2 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left({a}^{2}\right)\right)\right)\right) \cdot \pi\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left({a}^{2}\right)\right)\right) \cdot \pi\right) \]

      17. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left(a \cdot a\right)\right)\right) \cdot \pi\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      20. lower-neg.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 2.40000000000000018e211 < angle

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

   5. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 58.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 58.2

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

   7. Applied rewrites 58.2%

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

Alternative 6: 66.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(a\_m + b\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.6 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;angle\_m \leq 2.4 \cdot 10^{+211}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \mathsf{fma}\left(b, b, \left(-a\_m\right) \cdot a\_m\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (sin (* 0.011111111111111112 (* angle_m PI))) (- b a_m))
          (+ a_m b))))
   (*
    angle_s
    (if (<= angle_m 4.6e+16)
      t_0
      (if (<= angle_m 2.4e+211)
        (* 0.011111111111111112 (* (* angle_m (fma b b (* (- a_m) a_m))) PI))
        t_0)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * (b - a_m)) * (a_m + b);
	double tmp;
	if (angle_m <= 4.6e+16) {
		tmp = t_0;
	} else if (angle_m <= 2.4e+211) {
		tmp = 0.011111111111111112 * ((angle_m * fma(b, b, (-a_m * a_m))) * ((double) M_PI));
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * Float64(b - a_m)) * Float64(a_m + b))
	tmp = 0.0
	if (angle_m <= 4.6e+16)
		tmp = t_0;
	elseif (angle_m <= 2.4e+211)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * fma(b, b, Float64(Float64(-a_m) * a_m))) * pi));
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 4.6e+16], t$95$0, If[LessEqual[angle$95$m, 2.4e+211], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b * b + N[((-a$95$m) * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.6e16 or 2.40000000000000018e211 < angle

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 57.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 57.9

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 58.2

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

   3. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.9%

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

   if 4.6e16 < angle < 2.40000000000000018e211

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. lift-+.f64 N/A

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

      14. lift--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 54.9

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

   6. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      13. unpow2 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left({a}^{2}\right)\right)\right)\right) \cdot \pi\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left({a}^{2}\right)\right)\right) \cdot \pi\right) \]

      17. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left(a \cdot a\right)\right)\right) \cdot \pi\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      20. lower-neg.f64 53.2

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

   8. Applied rewrites 53.2%

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

Alternative 7: 66.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a\_m + b\right)\right) \cdot 2\right) \cdot \left(\left(b - a\_m\right) \cdot \cos \left(\left(-0.005555555555555556 \cdot \pi\right) \cdot angle\_m\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a_m b)) 2.0)
   (* (- b a_m) (cos (* (* -0.005555555555555556 PI) angle_m))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a_m + b)) * 2.0) * ((b - a_m) * cos(((-0.005555555555555556 * ((double) M_PI)) * angle_m))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((Math.sin(((0.005555555555555556 * angle_m) * Math.PI)) * (a_m + b)) * 2.0) * ((b - a_m) * Math.cos(((-0.005555555555555556 * Math.PI) * angle_m))));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (((math.sin(((0.005555555555555556 * angle_m) * math.pi)) * (a_m + b)) * 2.0) * ((b - a_m) * math.cos(((-0.005555555555555556 * math.pi) * angle_m))))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a_m + b)) * 2.0) * Float64(Float64(b - a_m) * cos(Float64(Float64(-0.005555555555555556 * pi) * angle_m)))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (((sin(((0.005555555555555556 * angle_m) * pi)) * (a_m + b)) * 2.0) * ((b - a_m) * cos(((-0.005555555555555556 * pi) * angle_m))));
end

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. difference-of-squares N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-*.f64 57.9

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 57.9

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

   19. lift-/.f64 N/A

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

   20. mult-flip N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 N/A

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

   23. metadata-eval 58.2

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

3. Applied rewrites 58.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 67.9

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 67.9

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

5. Applied rewrites 67.6%

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

7. Applied rewrites 67.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 67.9

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

9. Applied rewrites 67.9%

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

Alternative 8: 66.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a\_m + b\right)\right) \cdot 2\right) \cdot \left(\left(b - a\_m\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a_m b)) 2.0)
   (* (- b a_m) (cos (* -0.005555555555555556 (* angle_m PI)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a_m + b)) * 2.0) * ((b - a_m) * cos((-0.005555555555555556 * (angle_m * ((double) M_PI))))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((Math.sin(((0.005555555555555556 * angle_m) * Math.PI)) * (a_m + b)) * 2.0) * ((b - a_m) * Math.cos((-0.005555555555555556 * (angle_m * Math.PI)))));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (((math.sin(((0.005555555555555556 * angle_m) * math.pi)) * (a_m + b)) * 2.0) * ((b - a_m) * math.cos((-0.005555555555555556 * (angle_m * math.pi)))))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a_m + b)) * 2.0) * Float64(Float64(b - a_m) * cos(Float64(-0.005555555555555556 * Float64(angle_m * pi))))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (((sin(((0.005555555555555556 * angle_m) * pi)) * (a_m + b)) * 2.0) * ((b - a_m) * cos((-0.005555555555555556 * (angle_m * pi)))));
end

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. difference-of-squares N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-*.f64 57.9

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 57.9

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

   19. lift-/.f64 N/A

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

   20. mult-flip N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 N/A

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

   23. metadata-eval 58.2

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

3. Applied rewrites 58.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 67.9

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 67.9

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

5. Applied rewrites 67.6%

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

7. Applied rewrites 67.6%

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

Alternative 9: 66.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b - a\_m\right) \cdot \left(\sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \left(\pi + \pi\right)\right) \cdot \left(b + a\_m\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (- b a_m)
   (* (sin (* (* angle_m 0.005555555555555556) (+ PI PI))) (+ b a_m)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b - a_m) * (sin(((angle_m * 0.005555555555555556) * (((double) M_PI) + ((double) M_PI)))) * (b + a_m)));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * ((b - a_m) * (Math.sin(((angle_m * 0.005555555555555556) * (Math.PI + Math.PI))) * (b + a_m)));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * ((b - a_m) * (math.sin(((angle_m * 0.005555555555555556) * (math.pi + math.pi))) * (b + a_m)))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(b - a_m) * Float64(sin(Float64(Float64(angle_m * 0.005555555555555556) * Float64(pi + pi))) * Float64(b + a_m))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * ((b - a_m) * (sin(((angle_m * 0.005555555555555556) * (pi + pi))) * (b + a_m)));
end

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. difference-of-squares N/A

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

   12. lower-*.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-*.f64 57.9

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 57.9

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

   19. lift-/.f64 N/A

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

   20. mult-flip N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 N/A

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

   23. metadata-eval 58.2

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

3. Applied rewrites 58.2%

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

5. Applied rewrites 58.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right) \]

   12. pow1/3 N/A

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

   13. lift-PI.f64 N/A

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

   14. pow1/3 N/A

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

   15. pow-prod-up N/A

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

   16. lower-pow.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 N/A

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \frac{angle}{180}\right)\right)\right) \]

   20. lower-cbrt.f64 57.8

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \frac{angle}{180}\right)\right)\right) \]

   21. lift-/.f64 N/A

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\frac{angle}{180}}\right)\right)\right) \]

   22. mult-flip N/A

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)\right) \]

   24. *-commutative N/A

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right)\right)\right) \]

   25. lift-*.f64 57.8

       \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

7. Applied rewrites 57.8%

   \[\leadsto \left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right)} \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\sin \left(2 \cdot \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \cdot \left(a + b\right)\right)} \]

   6. lower-*.f64 67.5

      \[\leadsto \left(b - a\right) \cdot \color{blue}{\left(\sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a + b\right)\right)} \]

9. Applied rewrites 67.9%

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

Alternative 10: 63.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(angle\_m \cdot \left(a\_m + b\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \mathsf{fma}\left(b, b, \left(-a\_m\right) \cdot a\_m\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9e-17)
    (* (* (* angle_m (+ a_m b)) (- b a_m)) (* PI 0.011111111111111112))
    (* 0.011111111111111112 (* (* angle_m (fma b b (* (- a_m) a_m))) PI)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9e-17) {
		tmp = ((angle_m * (a_m + b)) * (b - a_m)) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * ((angle_m * fma(b, b, (-a_m * a_m))) * ((double) M_PI));
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 9e-17)
		tmp = Float64(Float64(Float64(angle_m * Float64(a_m + b)) * Float64(b - a_m)) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * fma(b, b, Float64(Float64(-a_m) * a_m))) * pi));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9e-17], N[(N[(N[(angle$95$m * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b * b + N[((-a$95$m) * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 8.99999999999999957e-17

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. lift-+.f64 N/A

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

      14. lift--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 54.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 54.9

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

   6. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if 8.99999999999999957e-17 < angle

   1. Initial program 54.1%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      16. lower-*.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. lower-+.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.9%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      8. difference-of-squares-rev N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - a \cdot a\right)\right) \cdot \pi\right) \]

      12. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left(a \cdot a\right)\right)\right)\right) \cdot \pi\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b + \left(\mathsf{neg}\left({a}^{2}\right)\right)\right)\right) \cdot \pi\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left({a}^{2}\right)\right)\right) \cdot \pi\right) \]

      17. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \mathsf{neg}\left(a \cdot a\right)\right)\right) \cdot \pi\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(a\right)\right) \cdot a\right)\right) \cdot \pi\right) \]

      20. lower-neg.f64 53.2

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)\right) \cdot \pi\right) \]

   8. Applied rewrites 53.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;\left(\left(angle\_m \cdot \left(a\_m + b\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(\left(-1 \cdot a\_m\right) \cdot \left(a\_m + b\right)\right)\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 6.8e+59)
    (* (* (* angle_m (+ a_m b)) (- b a_m)) (* PI 0.011111111111111112))
    (* 0.011111111111111112 (* (* angle_m (* (* -1.0 a_m) (+ a_m b))) PI)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 6.8e+59) {
		tmp = ((angle_m * (a_m + b)) * (b - a_m)) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * ((angle_m * ((-1.0 * a_m) * (a_m + b))) * ((double) M_PI));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 6.8e+59) {
		tmp = ((angle_m * (a_m + b)) * (b - a_m)) * (Math.PI * 0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * ((angle_m * ((-1.0 * a_m) * (a_m + b))) * Math.PI);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 6.8e+59:
		tmp = ((angle_m * (a_m + b)) * (b - a_m)) * (math.pi * 0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * ((angle_m * ((-1.0 * a_m) * (a_m + b))) * math.pi)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 6.8e+59)
		tmp = Float64(Float64(Float64(angle_m * Float64(a_m + b)) * Float64(b - a_m)) * Float64(pi * 0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(Float64(-1.0 * a_m) * Float64(a_m + b))) * pi));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 6.8e+59)
		tmp = ((angle_m * (a_m + b)) * (b - a_m)) * (pi * 0.011111111111111112);
	else
		tmp = 0.011111111111111112 * ((angle_m * ((-1.0 * a_m) * (a_m + b))) * pi);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 6.8e+59], N[(N[(N[(angle$95$m * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[(-1.0 * a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 6.80000000000000012e59

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      16. lower-*.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. lower-+.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.9%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

      4. associate-*l* N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

      12. lower-*.f64 62.9

          \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \color{blue}{0.011111111111111112}\right) \]

   8. Applied rewrites 62.9%

      \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\pi \cdot 0.011111111111111112\right)} \]

   if 6.80000000000000012e59 < angle

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      16. lower-*.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. lower-+.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.9%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(-1 \cdot a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 37.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(-1 \cdot a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   9. Applied rewrites 37.3%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(-1 \cdot a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(angle\_m \cdot \left(a\_m + b\right)\right) \cdot \left(b - a\_m\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (* (* (* angle_m (+ a_m b)) (- b a_m)) (* PI 0.011111111111111112))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((angle_m * (a_m + b)) * (b - a_m)) * (((double) M_PI) * 0.011111111111111112));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((angle_m * (a_m + b)) * (b - a_m)) * (Math.PI * 0.011111111111111112));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (((angle_m * (a_m + b)) * (b - a_m)) * (math.pi * 0.011111111111111112))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(angle_m * Float64(a_m + b)) * Float64(b - a_m)) * Float64(pi * 0.011111111111111112)))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (((angle_m * (a_m + b)) * (b - a_m)) * (pi * 0.011111111111111112));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(angle$95$m * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

   4. associate-*l* N/A

      \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

   9. associate-*r* N/A

      \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

   12. lower-*.f64 62.9

       \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \color{blue}{0.011111111111111112}\right) \]

8. Applied rewrites 62.9%

   \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\pi \cdot 0.011111111111111112\right)} \]
9. Add Preprocessing

Alternative 13: 62.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b - a\_m\right)\right) \cdot \left(\left(a\_m + b\right) \cdot \pi\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (* 0.011111111111111112 (* (* angle_m (- b a_m)) (* (+ a_m b) PI)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * (b - a_m)) * ((a_m + b) * ((double) M_PI))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * (b - a_m)) * ((a_m + b) * Math.PI)));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (0.011111111111111112 * ((angle_m * (b - a_m)) * ((a_m + b) * math.pi)))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b - a_m)) * Float64(Float64(a_m + b) * pi))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((angle_m * (b - a_m)) * ((a_m + b) * pi)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(a + b\right)\right) \cdot \pi\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \pi\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \pi\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \pi\right)\right) \]

   8. lower-*.f64 62.8

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(a + b\right) \cdot \color{blue}{\pi}\right)\right) \]

8. Applied rewrites 62.8%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \pi\right)}\right) \]
9. Add Preprocessing

Alternative 14: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a\_m}^{2}\right) \leq 10^{+60}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot \left(b + a\_m\right)\right) \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a_m 2.0))) 1e+60)
    (* 0.011111111111111112 (* (* angle_m (* (- b a_m) (+ a_m b))) PI))
    (* (* (* angle_m (+ b a_m)) b) (* PI 0.011111111111111112)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a_m, 2.0))) <= 1e+60) {
		tmp = 0.011111111111111112 * ((angle_m * ((b - a_m) * (a_m + b))) * ((double) M_PI));
	} else {
		tmp = ((angle_m * (b + a_m)) * b) * (((double) M_PI) * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0))) <= 1e+60) {
		tmp = 0.011111111111111112 * ((angle_m * ((b - a_m) * (a_m + b))) * Math.PI);
	} else {
		tmp = ((angle_m * (b + a_m)) * b) * (Math.PI * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))) <= 1e+60:
		tmp = 0.011111111111111112 * ((angle_m * ((b - a_m) * (a_m + b))) * math.pi)
	else:
		tmp = ((angle_m * (b + a_m)) * b) * (math.pi * 0.011111111111111112)
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0))) <= 1e+60)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(Float64(b - a_m) * Float64(a_m + b))) * pi));
	else
		tmp = Float64(Float64(Float64(angle_m * Float64(b + a_m)) * b) * Float64(pi * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a_m ^ 2.0))) <= 1e+60)
		tmp = 0.011111111111111112 * ((angle_m * ((b - a_m) * (a_m + b))) * pi);
	else
		tmp = ((angle_m * (b + a_m)) * b) * (pi * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+60], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle$95$m * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(Pi * 0.011111111111111112), $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)))) < 9.9999999999999995e59

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      16. lower-*.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. lower-+.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.9%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   if 9.9999999999999995e59 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 50.9

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      16. lower-*.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      18. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. lower-+.f64 54.9

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 54.9%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 38.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

       4. associate-*l* N/A

          \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       8. *-commutative N/A

          \[\leadsto \left(angle \cdot \left(\left(a + b\right) \cdot b\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       9. associate-*r* N/A

          \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       12. lift-+.f64 N/A

           \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       13. +-commutative N/A

           \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       14. lift-+.f64 N/A

           \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

       15. lower-*.f64 42.4

           \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \left(\pi \cdot \color{blue}{0.011111111111111112}\right) \]

   11. Applied rewrites 42.4%

       \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \color{blue}{\left(\pi \cdot 0.011111111111111112\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 42.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(angle\_m \cdot \left(b + a\_m\right)\right) \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* (* (* angle_m (+ b a_m)) b) (* PI 0.011111111111111112))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((angle_m * (b + a_m)) * b) * (((double) M_PI) * 0.011111111111111112));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((angle_m * (b + a_m)) * b) * (Math.PI * 0.011111111111111112));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (((angle_m * (b + a_m)) * b) * (math.pi * 0.011111111111111112))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(angle_m * Float64(b + a_m)) * b) * Float64(pi * 0.011111111111111112)))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (((angle_m * (b + a_m)) * b) * (pi * 0.011111111111111112));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(angle$95$m * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right)} \]

    2. *-commutative N/A

       \[\leadsto \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

    4. associate-*l* N/A

       \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{90}\right)} \]

    6. lift-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

    7. lift-*.f64 N/A

       \[\leadsto \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

    8. *-commutative N/A

       \[\leadsto \left(angle \cdot \left(\left(a + b\right) \cdot b\right)\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

    9. associate-*r* N/A

       \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\color{blue}{\pi} \cdot \frac{1}{90}\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

    12. lift-+.f64 N/A

        \[\leadsto \left(\left(angle \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

    13. +-commutative N/A

        \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

    14. lift-+.f64 N/A

        \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \left(\pi \cdot \frac{1}{90}\right) \]

    15. lower-*.f64 42.4

        \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \left(\pi \cdot \color{blue}{0.011111111111111112}\right) \]

11. Applied rewrites 42.4%

    \[\leadsto \left(\left(angle \cdot \left(b + a\right)\right) \cdot b\right) \cdot \color{blue}{\left(\pi \cdot 0.011111111111111112\right)} \]
12. Add Preprocessing

Alternative 16: 41.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(0.011111111111111112 \cdot \left(b \cdot angle\_m\right)\right) \cdot \left(b + a\_m\right)\right) \cdot \pi\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* (* (* 0.011111111111111112 (* b angle_m)) (+ b a_m)) PI)))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((0.011111111111111112 * (b * angle_m)) * (b + a_m)) * ((double) M_PI));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (((0.011111111111111112 * (b * angle_m)) * (b + a_m)) * Math.PI);
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (((0.011111111111111112 * (b * angle_m)) * (b + a_m)) * math.pi)

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(0.011111111111111112 * Float64(b * angle_m)) * Float64(b + a_m)) * pi))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (((0.011111111111111112 * (b * angle_m)) * (b + a_m)) * pi);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(0.011111111111111112 * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

    3. associate-*r* N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right)\right) \cdot \color{blue}{\pi} \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right)\right) \cdot \pi \]

    6. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right)\right) \cdot \pi \]

    7. associate-*r* N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(\left(angle \cdot b\right) \cdot \left(a + b\right)\right)\right) \cdot \pi \]

    8. associate-*r* N/A

       \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot b\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot b\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot b\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

    11. *-commutative N/A

        \[\leadsto \left(\left(\frac{1}{90} \cdot \left(b \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

    12. lower-*.f64 41.0

        \[\leadsto \left(\left(0.011111111111111112 \cdot \left(b \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

    13. lift-+.f64 N/A

        \[\leadsto \left(\left(\frac{1}{90} \cdot \left(b \cdot angle\right)\right) \cdot \left(a + b\right)\right) \cdot \pi \]

    14. +-commutative N/A

        \[\leadsto \left(\left(\frac{1}{90} \cdot \left(b \cdot angle\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

    15. lift-+.f64 41.0

        \[\leadsto \left(\left(0.011111111111111112 \cdot \left(b \cdot angle\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

11. Applied rewrites 41.0%

    \[\leadsto \left(\left(0.011111111111111112 \cdot \left(b \cdot angle\right)\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\pi} \]
12. Add Preprocessing

Alternative 17: 38.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(angle\_m \cdot \left(\left(b \cdot \left(\left(b + a\_m\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* angle_m (* (* b (* (+ b a_m) PI)) 0.011111111111111112))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (angle_m * ((b * ((b + a_m) * ((double) M_PI))) * 0.011111111111111112));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (angle_m * ((b * ((b + a_m) * Math.PI)) * 0.011111111111111112));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (angle_m * ((b * ((b + a_m) * math.pi)) * 0.011111111111111112))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(angle_m * Float64(Float64(b * Float64(Float64(b + a_m) * pi)) * 0.011111111111111112)))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (angle_m * ((b * ((b + a_m) * pi)) * 0.011111111111111112));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(angle$95$m * N[(N[(b * N[(N[(b + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right)} \]

    2. *-commutative N/A

       \[\leadsto \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \cdot \frac{1}{90} \]

    5. associate-*l* N/A

       \[\leadsto \left(angle \cdot \left(\left(b \cdot \left(a + b\right)\right) \cdot \pi\right)\right) \cdot \frac{1}{90} \]

    6. associate-*l* N/A

       \[\leadsto angle \cdot \color{blue}{\left(\left(\left(b \cdot \left(a + b\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto angle \cdot \color{blue}{\left(\left(\left(b \cdot \left(a + b\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto angle \cdot \left(\left(\left(b \cdot \left(a + b\right)\right) \cdot \pi\right) \cdot \color{blue}{\frac{1}{90}}\right) \]

    9. lift-*.f64 N/A

       \[\leadsto angle \cdot \left(\left(\left(b \cdot \left(a + b\right)\right) \cdot \pi\right) \cdot \frac{1}{90}\right) \]

    10. associate-*l* N/A

        \[\leadsto angle \cdot \left(\left(b \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot \frac{1}{90}\right) \]

    11. lower-*.f64 N/A

        \[\leadsto angle \cdot \left(\left(b \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot \frac{1}{90}\right) \]

    12. lower-*.f64 38.2

        \[\leadsto angle \cdot \left(\left(b \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \]

    13. lift-+.f64 N/A

        \[\leadsto angle \cdot \left(\left(b \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot \frac{1}{90}\right) \]

    14. +-commutative N/A

        \[\leadsto angle \cdot \left(\left(b \cdot \left(\left(b + a\right) \cdot \pi\right)\right) \cdot \frac{1}{90}\right) \]

    15. lift-+.f64 38.2

        \[\leadsto angle \cdot \left(\left(b \cdot \left(\left(b + a\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right) \]

11. Applied rewrites 38.2%

    \[\leadsto angle \cdot \color{blue}{\left(\left(b \cdot \left(\left(b + a\right) \cdot \pi\right)\right) \cdot 0.011111111111111112\right)} \]
12. Add Preprocessing

Alternative 18: 38.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(\pi \cdot angle\_m\right) \cdot \left(b \cdot \left(b + a\_m\right)\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* PI angle_m) (* b (+ b a_m))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((((double) M_PI) * angle_m) * (b * (b + a_m))));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((Math.PI * angle_m) * (b * (b + a_m))));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (0.011111111111111112 * ((math.pi * angle_m) * (b * (b + a_m))))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(pi * angle_m) * Float64(b * Float64(b + a_m)))))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((pi * angle_m) * (b * (b + a_m))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b * N[(b + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\pi \cdot \color{blue}{\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right)}\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(b \cdot \left(a + b\right)\right)}\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(b \cdot \left(a + b\right)\right)}\right) \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b} \cdot \left(a + b\right)\right)\right) \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b} \cdot \left(a + b\right)\right)\right) \]

    7. lower-*.f64 38.2

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\left(b \cdot \left(a + b\right)\right)}\right) \]

    8. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\color{blue}{b} \cdot \left(a + b\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\color{blue}{b} \cdot \left(a + b\right)\right)\right) \]

    10. lower-*.f64 38.2

        \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\color{blue}{b} \cdot \left(a + b\right)\right)\right) \]

    11. lift-+.f64 N/A

        \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(a + \color{blue}{b}\right)\right)\right) \]

    12. +-commutative N/A

        \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

    13. lift-+.f64 38.2

        \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b + \color{blue}{a}\right)\right)\right) \]

11. Applied rewrites 38.2%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{\left(b \cdot \left(b + a\right)\right)}\right) \]
12. Add Preprocessing

Alternative 19: 38.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b \cdot \left(a\_m + b\right)\right)\right) \cdot \pi\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* angle_m (* b (+ a_m b))) PI))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * (b * (a_m + b))) * ((double) M_PI)));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * (b * (a_m + b))) * Math.PI));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (0.011111111111111112 * ((angle_m * (b * (a_m + b))) * math.pi))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b * Float64(a_m + b))) * pi)))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((angle_m * (b * (a_m + b))) * pi));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
10. Add Preprocessing

Alternative 20: 36.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(b \cdot b\right)\right) \cdot \pi\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* angle_m (* b b)) PI))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * (b * b)) * ((double) M_PI)));
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((angle_m * (b * b)) * Math.PI));
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	return angle_s * (0.011111111111111112 * ((angle_m * (b * b)) * math.pi))

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(b * b)) * pi)))
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a_m, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((angle_m * (b * b)) * pi));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(angle$95$m * N[(b * b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 50.9

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   16. lower-*.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   18. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. lower-+.f64 54.9

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 54.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
8. Step-by-step derivation

9. Applied rewrites 38.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

10. Taylor expanded in a around 0

    \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot \pi\right) \]
11. Step-by-step derivation

12. Applied rewrites 36.3%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot \pi\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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)))))