ab-angle->ABCF B

Percentage Accurate: 53.4% → 67.2%

Time: 6.6s

Alternatives: 17

Speedup: 6.6×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% 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: 67.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.65 \cdot 10^{+138}:\\ \;\;\;\;\frac{\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)}{\frac{0.5}{b - a}} \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \pi \cdot 0.5\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.65e+138)
    (*
     (/
      (*
       (cos (* -0.005555555555555556 (* PI angle_m)))
       (sin (* (* 0.005555555555555556 angle_m) PI)))
      (/ 0.5 (- b a)))
     (+ a b))
    (*
     (- b a)
     (*
      (+ a b)
      (*
       (sin (fma (* 0.005555555555555556 angle_m) PI (* PI 0.5)))
       (* (sin (* 0.005555555555555556 (* angle_m PI))) 2.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.65e+138) {
		tmp = ((cos((-0.005555555555555556 * (((double) M_PI) * angle_m))) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)))) / (0.5 / (b - a))) * (a + b);
	} else {
		tmp = (b - a) * ((a + b) * (sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) * 0.5))) * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * 2.0)));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.65e+138)
		tmp = Float64(Float64(Float64(cos(Float64(-0.005555555555555556 * Float64(pi * angle_m))) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))) / Float64(0.5 / Float64(b - a))) * Float64(a + b));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(a + b) * Float64(sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi * 0.5))) * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * 2.0))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.65e+138], N[(N[(N[(N[Cos[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.64999999999999989e138

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\frac{\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{\frac{0.5}{b - a}} \cdot \left(a + b\right)} \]

   if 1.64999999999999989e138 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around inf

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 67.1

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

   8. Applied rewrites 67.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-sin.f64 N/A

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

      14. lift-*.f64 N/A

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

   10. Applied rewrites 67.1%

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

Alternative 2: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{+110}:\\ \;\;\;\;\frac{\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)}{\frac{0.5}{b - a}} \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(1 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9e+110)
    (*
     (/
      (*
       (cos (* -0.005555555555555556 (* PI angle_m)))
       (sin (* (* 0.005555555555555556 angle_m) PI)))
      (/ 0.5 (- b a)))
     (+ a b))
    (*
     (- b a)
     (*
      (+ a b)
      (* 1.0 (* (sin (* 0.005555555555555556 (* angle_m PI))) 2.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9e+110) {
		tmp = ((cos((-0.005555555555555556 * (((double) M_PI) * angle_m))) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)))) / (0.5 / (b - a))) * (a + b);
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9e+110) {
		tmp = ((Math.cos((-0.005555555555555556 * (Math.PI * angle_m))) * Math.sin(((0.005555555555555556 * angle_m) * Math.PI))) / (0.5 / (b - a))) * (a + b);
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * 2.0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 9e+110:
		tmp = ((math.cos((-0.005555555555555556 * (math.pi * angle_m))) * math.sin(((0.005555555555555556 * angle_m) * math.pi))) / (0.5 / (b - a))) * (a + b)
	else:
		tmp = (b - a) * ((a + b) * (1.0 * (math.sin((0.005555555555555556 * (angle_m * math.pi))) * 2.0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 9e+110)
		tmp = Float64(Float64(Float64(cos(Float64(-0.005555555555555556 * Float64(pi * angle_m))) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))) / Float64(0.5 / Float64(b - a))) * Float64(a + b));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(a + b) * Float64(1.0 * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * 2.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 9e+110)
		tmp = ((cos((-0.005555555555555556 * (pi * angle_m))) * sin(((0.005555555555555556 * angle_m) * pi))) / (0.5 / (b - a))) * (a + b);
	else
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * pi))) * 2.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9e+110], N[(N[(N[(N[Cos[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(1.0 * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.0000000000000005e110

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\frac{\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{\frac{0.5}{b - a}} \cdot \left(a + b\right)} \]

   if 9.0000000000000005e110 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around inf

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 67.1

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 65.6%

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

Alternative 3: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{+110}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \left(2 \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(1 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9e+110)
    (*
     (- b a)
     (*
      (*
       (cos (* -0.005555555555555556 (* PI angle_m)))
       (* 2.0 (sin (* (* 0.005555555555555556 angle_m) PI))))
      (+ a b)))
    (*
     (- b a)
     (*
      (+ a b)
      (* 1.0 (* (sin (* 0.005555555555555556 (* angle_m PI))) 2.0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9e+110) {
		tmp = (b - a) * ((cos((-0.005555555555555556 * (((double) M_PI) * angle_m))) * (2.0 * sin(((0.005555555555555556 * angle_m) * ((double) M_PI))))) * (a + b));
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9e+110) {
		tmp = (b - a) * ((Math.cos((-0.005555555555555556 * (Math.PI * angle_m))) * (2.0 * Math.sin(((0.005555555555555556 * angle_m) * Math.PI)))) * (a + b));
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * 2.0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 9e+110:
		tmp = (b - a) * ((math.cos((-0.005555555555555556 * (math.pi * angle_m))) * (2.0 * math.sin(((0.005555555555555556 * angle_m) * math.pi)))) * (a + b))
	else:
		tmp = (b - a) * ((a + b) * (1.0 * (math.sin((0.005555555555555556 * (angle_m * math.pi))) * 2.0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 9e+110)
		tmp = Float64(Float64(b - a) * Float64(Float64(cos(Float64(-0.005555555555555556 * Float64(pi * angle_m))) * Float64(2.0 * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)))) * Float64(a + b)));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(a + b) * Float64(1.0 * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * 2.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 9e+110)
		tmp = (b - a) * ((cos((-0.005555555555555556 * (pi * angle_m))) * (2.0 * sin(((0.005555555555555556 * angle_m) * pi)))) * (a + b));
	else
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * pi))) * 2.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9e+110], N[(N[(b - a), $MachinePrecision] * N[(N[(N[Cos[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(1.0 * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.0000000000000005e110

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

   7. Applied rewrites 66.9%

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

   if 9.0000000000000005e110 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around inf

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 67.1

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 65.6%

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

Alternative 4: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1900:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 9.6 \cdot 10^{+110}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b \cdot \cos \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right) \cdot \left(2 \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(1 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1900.0)
    (* (- b a) (* 0.011111111111111112 (* angle_m (* PI (+ a b)))))
    (if (<= angle_m 9.6e+110)
      (*
       (* (- b a) (* b (cos (* -0.005555555555555556 (* angle_m PI)))))
       (* 2.0 (sin (* (* 0.005555555555555556 angle_m) PI))))
      (*
       (- b a)
       (*
        (+ a b)
        (* 1.0 (* (sin (* 0.005555555555555556 (* angle_m PI))) 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1900.0) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (a + b))));
	} else if (angle_m <= 9.6e+110) {
		tmp = ((b - a) * (b * cos((-0.005555555555555556 * (angle_m * ((double) M_PI)))))) * (2.0 * sin(((0.005555555555555556 * angle_m) * ((double) M_PI))));
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1900.0) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (Math.PI * (a + b))));
	} else if (angle_m <= 9.6e+110) {
		tmp = ((b - a) * (b * Math.cos((-0.005555555555555556 * (angle_m * Math.PI))))) * (2.0 * Math.sin(((0.005555555555555556 * angle_m) * Math.PI)));
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * 2.0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 1900.0:
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (math.pi * (a + b))))
	elif angle_m <= 9.6e+110:
		tmp = ((b - a) * (b * math.cos((-0.005555555555555556 * (angle_m * math.pi))))) * (2.0 * math.sin(((0.005555555555555556 * angle_m) * math.pi)))
	else:
		tmp = (b - a) * ((a + b) * (1.0 * (math.sin((0.005555555555555556 * (angle_m * math.pi))) * 2.0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1900.0)
		tmp = Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a + b)))));
	elseif (angle_m <= 9.6e+110)
		tmp = Float64(Float64(Float64(b - a) * Float64(b * cos(Float64(-0.005555555555555556 * Float64(angle_m * pi))))) * Float64(2.0 * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(a + b) * Float64(1.0 * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * 2.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 1900.0)
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (pi * (a + b))));
	elseif (angle_m <= 9.6e+110)
		tmp = ((b - a) * (b * cos((-0.005555555555555556 * (angle_m * pi))))) * (2.0 * sin(((0.005555555555555556 * angle_m) * pi)));
	else
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * pi))) * 2.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1900.0], N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 9.6e+110], N[(N[(N[(b - a), $MachinePrecision] * N[(b * N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(1.0 * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 1900

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6

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

   8. Applied rewrites 62.6%

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

   if 1900 < angle < 9.60000000000000049e110

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 37.4

         \[\leadsto \left(\left(b - a\right) \cdot \left(b \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \left(2 \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \]

   10. Applied rewrites 37.4%

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

   if 9.60000000000000049e110 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around inf

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 67.1

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 65.6%

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

Alternative 5: 65.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{+110}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot 2\right) \cdot \left(\cos t\_0 \cdot \left(b - a\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(1 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle_m) PI)))
   (*
    angle_s
    (if (<= angle_m 9e+110)
      (* (* (* (sin t_0) 2.0) (* (cos t_0) (- b a))) (+ a b))
      (*
       (- b a)
       (*
        (+ a b)
        (* 1.0 (* (sin (* 0.005555555555555556 (* angle_m PI))) 2.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (0.005555555555555556 * angle_m) * ((double) M_PI);
	double tmp;
	if (angle_m <= 9e+110) {
		tmp = ((sin(t_0) * 2.0) * (cos(t_0) * (b - a))) * (a + b);
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * 2.0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (0.005555555555555556 * angle_m) * Math.PI;
	double tmp;
	if (angle_m <= 9e+110) {
		tmp = ((Math.sin(t_0) * 2.0) * (Math.cos(t_0) * (b - a))) * (a + b);
	} else {
		tmp = (b - a) * ((a + b) * (1.0 * (Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * 2.0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (0.005555555555555556 * angle_m) * math.pi
	tmp = 0
	if angle_m <= 9e+110:
		tmp = ((math.sin(t_0) * 2.0) * (math.cos(t_0) * (b - a))) * (a + b)
	else:
		tmp = (b - a) * ((a + b) * (1.0 * (math.sin((0.005555555555555556 * (angle_m * math.pi))) * 2.0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(0.005555555555555556 * angle_m) * pi)
	tmp = 0.0
	if (angle_m <= 9e+110)
		tmp = Float64(Float64(Float64(sin(t_0) * 2.0) * Float64(cos(t_0) * Float64(b - a))) * Float64(a + b));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(a + b) * Float64(1.0 * Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * 2.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (0.005555555555555556 * angle_m) * pi;
	tmp = 0.0;
	if (angle_m <= 9e+110)
		tmp = ((sin(t_0) * 2.0) * (cos(t_0) * (b - a))) * (a + b);
	else
		tmp = (b - a) * ((a + b) * (1.0 * (sin((0.005555555555555556 * (angle_m * pi))) * 2.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 9e+110], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(1.0 * N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.0000000000000005e110

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 56.5%

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

   9. Applied rewrites 67.0%

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

   if 9.0000000000000005e110 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around inf

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 67.1

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 65.6%

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

Alternative 6: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 10^{+259}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \left(1 \cdot \left(\sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right) \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         1e+259)
      (*
       (- b a)
       (*
        (+ a b)
        (* 1.0 (* (sin (* (* angle_m 0.005555555555555556) PI)) 2.0))))
      (* (- b a) (* 0.011111111111111112 (* angle_m (* PI (+ a b)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+259) {
		tmp = (b - a) * ((a + b) * (1.0 * (sin(((angle_m * 0.005555555555555556) * ((double) M_PI))) * 2.0)));
	} else {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (a + b))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 1e+259) {
		tmp = (b - a) * ((a + b) * (1.0 * (Math.sin(((angle_m * 0.005555555555555556) * Math.PI)) * 2.0)));
	} else {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (Math.PI * (a + b))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= 1e+259:
		tmp = (b - a) * ((a + b) * (1.0 * (math.sin(((angle_m * 0.005555555555555556) * math.pi)) * 2.0)))
	else:
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (math.pi * (a + b))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+259)
		tmp = Float64(Float64(b - a) * Float64(Float64(a + b) * Float64(1.0 * Float64(sin(Float64(Float64(angle_m * 0.005555555555555556) * pi)) * 2.0))));
	else
		tmp = Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a + b)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 1e+259)
		tmp = (b - a) * ((a + b) * (1.0 * (sin(((angle_m * 0.005555555555555556) * pi)) * 2.0)));
	else
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (pi * (a + b))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[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], 1e+259], N[(N[(b - a), $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(1.0 * N[(N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 9.999999999999999e258

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 65.7%

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

   if 9.999999999999999e258 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6

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

   8. Applied rewrites 62.6%

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

Alternative 7: 65.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.9 \cdot 10^{+35}:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 5.9e+35)
    (* (- b a) (* 0.011111111111111112 (* angle_m (* PI (+ a b)))))
    (*
     (* (+ a b) (- b a))
     (* 2.0 (sin (* (* 0.005555555555555556 angle_m) PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 5.9e+35) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (a + b))));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * sin(((0.005555555555555556 * angle_m) * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 5.9e+35) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (Math.PI * (a + b))));
	} else {
		tmp = ((a + b) * (b - a)) * (2.0 * Math.sin(((0.005555555555555556 * angle_m) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 5.9e+35:
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (math.pi * (a + b))))
	else:
		tmp = ((a + b) * (b - a)) * (2.0 * math.sin(((0.005555555555555556 * angle_m) * math.pi)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 5.9e+35)
		tmp = Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a + b)))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(b - a)) * Float64(2.0 * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 5.9e+35)
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (pi * (a + b))));
	else
		tmp = ((a + b) * (b - a)) * (2.0 * sin(((0.005555555555555556 * angle_m) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 5.9e+35], N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 5.89999999999999985e35

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6

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

   8. Applied rewrites 62.6%

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

   if 5.89999999999999985e35 < angle

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 55.3

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

   10. Applied rewrites 55.3%

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

Alternative 8: 64.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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 56.8

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

   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 56.8

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

   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. div-flip N/A

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

   21. associate-/r/ 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 56.5

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

3. Applied rewrites 56.5%

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

5. Applied rewrites 67.0%

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

6. Taylor expanded in angle around inf

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

   1. lower-sin.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 67.1

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

8. Applied rewrites 67.1%

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

9. Taylor expanded in angle around 0

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

11. Applied rewrites 65.6%

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

Alternative 9: 64.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 7.8 \cdot 10^{+73}:\\ \;\;\;\;\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle\_m \cdot \frac{\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot b}{a - b}\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 7.8e+73)
    (* (- b a) (* 0.011111111111111112 (* angle_m (* PI (+ a b)))))
    (*
     0.011111111111111112
     (* (* angle_m (/ (* (* (+ a b) (- a b)) b) (- a b))) PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 7.8e+73) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (a + b))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * ((((a + b) * (a - b)) * b) / (a - b))) * ((double) M_PI));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 7.8e+73) {
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (Math.PI * (a + b))));
	} else {
		tmp = 0.011111111111111112 * ((angle_m * ((((a + b) * (a - b)) * b) / (a - b))) * Math.PI);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 7.8e+73:
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (math.pi * (a + b))))
	else:
		tmp = 0.011111111111111112 * ((angle_m * ((((a + b) * (a - b)) * b) / (a - b))) * math.pi)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 7.8e+73)
		tmp = Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(a + b)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(Float64(Float64(Float64(a + b) * Float64(a - b)) * b) / Float64(a - b))) * pi));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 7.8e+73)
		tmp = (b - a) * (0.011111111111111112 * (angle_m * (pi * (a + b))));
	else
		tmp = 0.011111111111111112 * ((angle_m * ((((a + b) * (a - b)) * b) / (a - b))) * pi);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 7.8e+73], N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[(N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 7.8000000000000002e73

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.8

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

      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 56.8

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

      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. div-flip N/A

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

      21. associate-/r/ 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 56.5

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

   3. Applied rewrites 56.5%

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-+.f64 62.6

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

   8. Applied rewrites 62.6%

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

   if 7.8000000000000002e73 < angle

   1. Initial program 53.4%

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

   2. 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 50.4

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.4

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

      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. pow2 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. pow2 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 53.8

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

      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 53.8

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

   6. Applied rewrites 53.8%

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

   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 37.0%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. flip-+ N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. difference-of-squares N/A

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

       9. lift-+.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 N/A

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

       12. lower--.f64 37.3

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

   11. Applied rewrites 37.3%

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

Alternative 10: 62.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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 56.8

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

   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 56.8

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

   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. div-flip N/A

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

   21. associate-/r/ 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 56.5

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

3. Applied rewrites 56.5%

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

5. Applied rewrites 67.0%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-+.f64 62.6

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

8. Applied rewrites 62.6%

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

Alternative 11: 62.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

2. 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 50.4

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

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.4

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

   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. pow2 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. pow2 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 53.8

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

   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 53.8

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

6. Applied rewrites 53.8%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
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 62.6

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

8. Applied rewrites 62.6%

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

Alternative 12: 62.5% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

2. 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 50.4

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

4. Applied rewrites 50.4%

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

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

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

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

   10. lift-+.f64 N/A

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

   11. lift--.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 62.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 62.5

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

6. Applied rewrites 62.5%

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

Alternative 13: 58.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\left(angle\_m \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right)\\ t_1 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(angle\_m \cdot \left(a + b\right)\right) \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.011111111111111112 (* (* angle_m (* (- b a) (+ a b))) PI)))
        (t_1 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_1 1e+160)
      t_0
      (if (<= t_1 INFINITY)
        (* (* (* angle_m (+ a b)) b) (* PI 0.011111111111111112))
        t_0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 0.011111111111111112 * ((angle_m * ((b - a) * (a + b))) * ((double) M_PI));
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_1 <= 1e+160) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((angle_m * (a + b)) * b) * (((double) M_PI) * 0.011111111111111112);
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 0.011111111111111112 * ((angle_m * ((b - a) * (a + b))) * Math.PI);
	double t_1 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if (t_1 <= 1e+160) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((angle_m * (a + b)) * b) * (Math.PI * 0.011111111111111112);
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = 0.011111111111111112 * ((angle_m * ((b - a) * (a + b))) * math.pi)
	t_1 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	tmp = 0
	if t_1 <= 1e+160:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((angle_m * (a + b)) * b) * (math.pi * 0.011111111111111112)
	else:
		tmp = t_0
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(0.011111111111111112 * Float64(Float64(angle_m * Float64(Float64(b - a) * Float64(a + b))) * pi))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_1 <= 1e+160)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(angle_m * Float64(a + b)) * b) * Float64(pi * 0.011111111111111112));
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = 0.011111111111111112 * ((angle_m * ((b - a) * (a + b))) * pi);
	t_1 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	tmp = 0.0;
	if (t_1 <= 1e+160)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((angle_m * (a + b)) * b) * (pi * 0.011111111111111112);
	else
		tmp = t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.011111111111111112 * N[(N[(angle$95$m * N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, 1e+160], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(angle$95$m * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.00000000000000001e160 or +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.4%

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

   2. 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 50.4

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.4

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

      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. pow2 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. pow2 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 53.8

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

      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 53.8

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

   6. Applied rewrites 53.8%

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

   if 1.00000000000000001e160 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 53.4%

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

   2. 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 50.4

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   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 50.4

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

      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. pow2 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. pow2 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 53.8

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

      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 53.8

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

   6. Applied rewrites 53.8%

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

   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 37.0%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
   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. lower-*.f64 41.3

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

   11. Applied rewrites 41.3%

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

Alternative 14: 41.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

2. 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 50.4

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

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.4

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

   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. pow2 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. pow2 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 53.8

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

   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 53.8

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

6. Applied rewrites 53.8%

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

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 37.0%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
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. lower-*.f64 41.3

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

11. Applied rewrites 41.3%

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

Alternative 15: 41.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

2. 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 50.4

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

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.4

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

   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. pow2 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. pow2 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 53.8

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

   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 53.8

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

6. Applied rewrites 53.8%

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

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 37.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 41.3

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

11. Applied rewrites 41.3%

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

Alternative 16: 40.1% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

2. 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 50.4

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

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.4

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

   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. pow2 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. pow2 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 53.8

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

   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 53.8

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

6. Applied rewrites 53.8%

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

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 37.0%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
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. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*r* N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lower-*.f64 40.1

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

11. Applied rewrites 40.1%

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

Alternative 17: 37.0% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

2. 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 50.4

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

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
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 50.4

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

   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. pow2 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. pow2 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 53.8

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

   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 53.8

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

6. Applied rewrites 53.8%

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

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 37.0%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(b \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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)))))