ab-angle->ABCF B

Percentage Accurate: 53.4% → 68.0%

Time: 6.0s

Alternatives: 18

Speedup: 6.6×

• 32.9% 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: 68.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 3.45 \cdot 10^{+174}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b_m))
          (- b_m a))))
   (*
    angle_s
    (if (<= b_m 3.45e+174)
      (*
       (* 2.0 (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))))
       t_0)
      (* 2.0 t_0)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b_m)) * (b_m - a);
	double tmp;
	if (b_m <= 3.45e+174) {
		tmp = (2.0 * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)))) * t_0;
	} else {
		tmp = 2.0 * t_0;
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b_m)) * Float64(b_m - a))
	tmp = 0.0
	if (b_m <= 3.45e+174)
		tmp = Float64(Float64(2.0 * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))) * t_0);
	else
		tmp = Float64(2.0 * t_0);
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 3.45e+174], N[(N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(2.0 * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.4500000000000001e174

   1. Initial program 57.5%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 65.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 65.0

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

   8. Applied rewrites 65.0%

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

   if 3.4500000000000001e174 < b

   1. Initial program 38.6%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 74.8%

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

Alternative 2: 67.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(\sin t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle_m) PI)))
   (*
    angle_s
    (if (<= angle_m 1.15e+62)
      (* (* 2.0 (cos t_0)) (* (* (sin t_0) (+ a b_m)) (- b_m a)))
      (*
       (* 2.0 (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))))
       (*
        (sin (* (* PI angle_m) 0.005555555555555556))
        (* (+ b_m a) (- b_m a))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (0.005555555555555556 * angle_m) * ((double) M_PI);
	double tmp;
	if (angle_m <= 1.15e+62) {
		tmp = (2.0 * cos(t_0)) * ((sin(t_0) * (a + b_m)) * (b_m - a));
	} else {
		tmp = (2.0 * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)))) * (sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * ((b_m + a) * (b_m - a)));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(0.005555555555555556 * angle_m) * pi)
	tmp = 0.0
	if (angle_m <= 1.15e+62)
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(sin(t_0) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(2.0 * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))) * Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * Float64(Float64(b_m + a) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, 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, 1.15e+62], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.14999999999999992e62

   1. Initial program 69.5%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 90.8

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

   8. Applied rewrites 90.8%

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

   if 1.14999999999999992e62 < angle

   1. Initial program 29.0%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 32.6

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

   6. Applied rewrites 32.6%

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

Alternative 3: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\\ t_1 := \left(\sin t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.8 \cdot 10^{+42}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle_m) PI))
        (t_1 (* (* (sin t_0) (+ a b_m)) (- b_m a))))
   (*
    angle_s
    (if (<= angle_m 3.8e+42) (* (* 2.0 (cos t_0)) t_1) (* 2.0 t_1)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (0.005555555555555556 * angle_m) * ((double) M_PI);
	double t_1 = (sin(t_0) * (a + b_m)) * (b_m - a);
	double tmp;
	if (angle_m <= 3.8e+42) {
		tmp = (2.0 * cos(t_0)) * t_1;
	} else {
		tmp = 2.0 * t_1;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (0.005555555555555556 * angle_m) * Math.PI;
	double t_1 = (Math.sin(t_0) * (a + b_m)) * (b_m - a);
	double tmp;
	if (angle_m <= 3.8e+42) {
		tmp = (2.0 * Math.cos(t_0)) * t_1;
	} else {
		tmp = 2.0 * t_1;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = (0.005555555555555556 * angle_m) * math.pi
	t_1 = (math.sin(t_0) * (a + b_m)) * (b_m - a)
	tmp = 0
	if angle_m <= 3.8e+42:
		tmp = (2.0 * math.cos(t_0)) * t_1
	else:
		tmp = 2.0 * t_1
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(0.005555555555555556 * angle_m) * pi)
	t_1 = Float64(Float64(sin(t_0) * Float64(a + b_m)) * Float64(b_m - a))
	tmp = 0.0
	if (angle_m <= 3.8e+42)
		tmp = Float64(Float64(2.0 * cos(t_0)) * t_1);
	else
		tmp = Float64(2.0 * t_1);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = (0.005555555555555556 * angle_m) * pi;
	t_1 = (sin(t_0) * (a + b_m)) * (b_m - a);
	tmp = 0.0;
	if (angle_m <= 3.8e+42)
		tmp = (2.0 * cos(t_0)) * t_1;
	else
		tmp = 2.0 * t_1;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 3.8e+42], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 * t$95$1), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.7999999999999998e42

   1. Initial program 71.5%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 94.0%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 93.9

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

   8. Applied rewrites 93.9%

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

   if 3.7999999999999998e42 < angle

   1. Initial program 29.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 32.1%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 33.5%

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

Alternative 4: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5.2 \cdot 10^{-192}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(t\_0 \cdot b\_m\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(t\_0 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* 0.005555555555555556 angle_m) PI))))
   (*
    angle_s
    (if (<= a 5.2e-192)
      (*
       (* 2.0 (cos (* (* PI angle_m) 0.005555555555555556)))
       (* (* t_0 b_m) (- b_m a)))
      (* 2.0 (* (* t_0 (+ a b_m)) (- b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = sin(((0.005555555555555556 * angle_m) * ((double) M_PI)));
	double tmp;
	if (a <= 5.2e-192) {
		tmp = (2.0 * cos(((((double) M_PI) * angle_m) * 0.005555555555555556))) * ((t_0 * b_m) * (b_m - a));
	} else {
		tmp = 2.0 * ((t_0 * (a + b_m)) * (b_m - a));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = Math.sin(((0.005555555555555556 * angle_m) * Math.PI));
	double tmp;
	if (a <= 5.2e-192) {
		tmp = (2.0 * Math.cos(((Math.PI * angle_m) * 0.005555555555555556))) * ((t_0 * b_m) * (b_m - a));
	} else {
		tmp = 2.0 * ((t_0 * (a + b_m)) * (b_m - a));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = math.sin(((0.005555555555555556 * angle_m) * math.pi))
	tmp = 0
	if a <= 5.2e-192:
		tmp = (2.0 * math.cos(((math.pi * angle_m) * 0.005555555555555556))) * ((t_0 * b_m) * (b_m - a))
	else:
		tmp = 2.0 * ((t_0 * (a + b_m)) * (b_m - a))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))
	tmp = 0.0
	if (a <= 5.2e-192)
		tmp = Float64(Float64(2.0 * cos(Float64(Float64(pi * angle_m) * 0.005555555555555556))) * Float64(Float64(t_0 * b_m) * Float64(b_m - a)));
	else
		tmp = Float64(2.0 * Float64(Float64(t_0 * Float64(a + b_m)) * Float64(b_m - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = sin(((0.005555555555555556 * angle_m) * pi));
	tmp = 0.0;
	if (a <= 5.2e-192)
		tmp = (2.0 * cos(((pi * angle_m) * 0.005555555555555556))) * ((t_0 * b_m) * (b_m - a));
	else
		tmp = 2.0 * ((t_0 * (a + b_m)) * (b_m - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 5.2e-192], N[(N[(2.0 * N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$0 * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.2000000000000003e-192

   1. Initial program 55.2%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 67.7%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 45.2%

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

   if 5.2000000000000003e-192 < a

   1. Initial program 50.7%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 65.1%

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

Alternative 5: 66.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 57.4%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift--.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

6. Applied rewrites 67.4%

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

Alternative 6: 66.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 57.4%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift--.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

6. Applied rewrites 67.4%

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

7. Taylor expanded in angle around 0

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

9. Applied rewrites 66.5%

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

Alternative 7: 62.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9 \cdot 10^{+41}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9e+41)
    (* (* (* (* angle_m PI) (+ a b_m)) (- b_m a)) 0.011111111111111112)
    (*
     2.0
     (*
      (sin (* (* PI angle_m) 0.005555555555555556))
      (* (+ b_m a) (- b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9e+41) {
		tmp = (((angle_m * ((double) M_PI)) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * ((b_m + a) * (b_m - a)));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9e+41) {
		tmp = (((angle_m * Math.PI) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else {
		tmp = 2.0 * (Math.sin(((Math.PI * angle_m) * 0.005555555555555556)) * ((b_m + a) * (b_m - a)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if angle_m <= 9e+41:
		tmp = (((angle_m * math.pi) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	else:
		tmp = 2.0 * (math.sin(((math.pi * angle_m) * 0.005555555555555556)) * ((b_m + a) * (b_m - a)))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 9e+41)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	else
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * Float64(Float64(b_m + a) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 9e+41)
		tmp = (((angle_m * pi) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	else
		tmp = 2.0 * (sin(((pi * angle_m) * 0.005555555555555556)) * ((b_m + a) * (b_m - a)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9e+41], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.0000000000000002e41

   1. Initial program 71.7%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 73.2

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

   4. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 91.1

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

   6. Applied rewrites 91.1%

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

   if 9.0000000000000002e41 < angle

   1. Initial program 29.2%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 33.2%

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

Alternative 8: 62.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.9 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;angle\_m \leq 6.1 \cdot 10^{+165}:\\ \;\;\;\;\log \left(e^{\pi \cdot \left(\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot angle\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \pi\right) \cdot \left(\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 5.9e+23)
    (* (* (* (* angle_m PI) (+ a b_m)) (- b_m a)) 0.011111111111111112)
    (if (<= angle_m 6.1e+165)
      (log (exp (* PI (* (* -0.011111111111111112 (* a a)) angle_m))))
      (* (* angle_m PI) (* (* (- b_m a) (+ a b_m)) 0.011111111111111112))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 5.9e+23) {
		tmp = (((angle_m * ((double) M_PI)) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else if (angle_m <= 6.1e+165) {
		tmp = log(exp((((double) M_PI) * ((-0.011111111111111112 * (a * a)) * angle_m))));
	} else {
		tmp = (angle_m * ((double) M_PI)) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 5.9e+23) {
		tmp = (((angle_m * Math.PI) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	} else if (angle_m <= 6.1e+165) {
		tmp = Math.log(Math.exp((Math.PI * ((-0.011111111111111112 * (a * a)) * angle_m))));
	} else {
		tmp = (angle_m * Math.PI) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if angle_m <= 5.9e+23:
		tmp = (((angle_m * math.pi) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	elif angle_m <= 6.1e+165:
		tmp = math.log(math.exp((math.pi * ((-0.011111111111111112 * (a * a)) * angle_m))))
	else:
		tmp = (angle_m * math.pi) * (((b_m - a) * (a + b_m)) * 0.011111111111111112)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 5.9e+23)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	elseif (angle_m <= 6.1e+165)
		tmp = log(exp(Float64(pi * Float64(Float64(-0.011111111111111112 * Float64(a * a)) * angle_m))));
	else
		tmp = Float64(Float64(angle_m * pi) * Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 5.9e+23)
		tmp = (((angle_m * pi) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	elseif (angle_m <= 6.1e+165)
		tmp = log(exp((pi * ((-0.011111111111111112 * (a * a)) * angle_m))));
	else
		tmp = (angle_m * pi) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 5.9e+23], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[angle$95$m, 6.1e+165], N[Log[N[Exp[N[(Pi * N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 5.89999999999999987e23

   1. Initial program 73.9%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 75.6

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

   4. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 94.4

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

   6. Applied rewrites 94.4%

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

   if 5.89999999999999987e23 < angle < 6.09999999999999965e165

   1. Initial program 28.7%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 20.8

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

   7. Applied rewrites 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. add-log-exp N/A

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

      6. log-pow-rev N/A

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

      7. lower-log.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 23.5

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

   9. Applied rewrites 23.5%

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

       1. lift-pow.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-exp.f64 N/A

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

       4. pow-exp N/A

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

       5. lower-exp.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 23.5

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

   11. Applied rewrites 23.5%

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

   if 6.09999999999999965e165 < angle

   1. Initial program 29.6%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 27.3

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

   4. Applied rewrites 27.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 27.4

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

   6. Applied rewrites 27.4%

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

Alternative 9: 62.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 7.5 \cdot 10^{-112}:\\ \;\;\;\;\left(\left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(a \cdot a\right)\right) \cdot -2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 7.5e-112)
    (* (* (* (sin (* (* PI angle_m) 0.005555555555555556)) (* a a)) -2.0) 1.0)
    (* (* (* (* angle_m PI) (+ a b_m)) (- b_m a)) 0.011111111111111112))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 7.5e-112) {
		tmp = ((sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * (a * a)) * -2.0) * 1.0;
	} else {
		tmp = (((angle_m * ((double) M_PI)) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 7.5e-112) {
		tmp = ((Math.sin(((Math.PI * angle_m) * 0.005555555555555556)) * (a * a)) * -2.0) * 1.0;
	} else {
		tmp = (((angle_m * Math.PI) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if b_m <= 7.5e-112:
		tmp = ((math.sin(((math.pi * angle_m) * 0.005555555555555556)) * (a * a)) * -2.0) * 1.0
	else:
		tmp = (((angle_m * math.pi) * (a + b_m)) * (b_m - a)) * 0.011111111111111112
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 7.5e-112)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * Float64(a * a)) * -2.0) * 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (b_m <= 7.5e-112)
		tmp = ((sin(((pi * angle_m) * 0.005555555555555556)) * (a * a)) * -2.0) * 1.0;
	else
		tmp = (((angle_m * pi) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 7.5e-112], N[(N[(N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 7.5000000000000002e-112

   1. Initial program 64.2%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 63.7

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

   4. Applied rewrites 63.7%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 61.7%

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

   if 7.5000000000000002e-112 < b

   1. Initial program 48.3%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 51.0

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

   4. Applied rewrites 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 63.5

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

   6. Applied rewrites 63.5%

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

Alternative 10: 58.1% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower--.f64 53.7

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

4. Applied rewrites 53.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lift--.f64 62.2

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

6. Applied rewrites 62.2%

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

Alternative 11: 57.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -8.15 \cdot 10^{-215}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(b\_m \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(a \cdot \left(b\_m - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b_m 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 -8.15e-215)
      (* (* (* (* angle_m PI) -0.011111111111111112) a) a)
      (if (<= t_0 INFINITY)
        (* (* (* PI angle_m) (* b_m b_m)) 0.011111111111111112)
        (* (* (* PI angle_m) (* a (- b_m a))) 0.011111111111111112))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = 2.0 * (pow(b_m, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -8.15e-215) {
		tmp = (((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((((double) M_PI) * angle_m) * (b_m * b_m)) * 0.011111111111111112;
	} else {
		tmp = ((((double) M_PI) * angle_m) * (a * (b_m - a))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = 2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if (t_0 <= -8.15e-215) {
		tmp = (((angle_m * Math.PI) * -0.011111111111111112) * a) * a;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.PI * angle_m) * (b_m * b_m)) * 0.011111111111111112;
	} else {
		tmp = ((Math.PI * angle_m) * (a * (b_m - a))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	t_0 = 2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))
	tmp = 0
	if t_0 <= -8.15e-215:
		tmp = (((angle_m * math.pi) * -0.011111111111111112) * a) * a
	elif t_0 <= math.inf:
		tmp = ((math.pi * angle_m) * (b_m * b_m)) * 0.011111111111111112
	else:
		tmp = ((math.pi * angle_m) * (a * (b_m - a))) * 0.011111111111111112
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -8.15e-215)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(b_m * b_m)) * 0.011111111111111112);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(a * Float64(b_m - a))) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	t_0 = 2.0 * ((b_m ^ 2.0) - (a ^ 2.0));
	tmp = 0.0;
	if (t_0 <= -8.15e-215)
		tmp = (((angle_m * pi) * -0.011111111111111112) * a) * a;
	elseif (t_0 <= Inf)
		tmp = ((pi * angle_m) * (b_m * b_m)) * 0.011111111111111112;
	else
		tmp = ((pi * angle_m) * (a * (b_m - a))) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -8.15e-215], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(a * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -8.15e-215

   1. Initial program 55.2%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot angle, \pi, -\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a}, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right)}{a}\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 50.9

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

   10. Applied rewrites 50.9%

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

       1. lift-*.f64 N/A

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

   12. Applied rewrites 61.7%

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

   if -8.15e-215 < (*.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 58.5%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 54.9

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot {b}^{2}\right) \cdot \frac{1}{90} \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. pow2 N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.7

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

   7. Applied rewrites 54.7%

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

   if +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 0.0%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in a around inf

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(a \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.5%

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

Alternative 12: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -2 \cdot 10^{-143}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot \pi\right) \cdot \left(\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -2e-143)
    (* (* (* (* angle_m PI) -0.011111111111111112) a) a)
    (* (* angle_m PI) (* (* (- b_m a) (+ a b_m)) 0.011111111111111112)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -2e-143) {
		tmp = (((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a;
	} else {
		tmp = (angle_m * ((double) M_PI)) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -2e-143) {
		tmp = (((angle_m * Math.PI) * -0.011111111111111112) * a) * a;
	} else {
		tmp = (angle_m * Math.PI) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -2e-143:
		tmp = (((angle_m * math.pi) * -0.011111111111111112) * a) * a
	else:
		tmp = (angle_m * math.pi) * (((b_m - a) * (a + b_m)) * 0.011111111111111112)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-143)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a);
	else
		tmp = Float64(Float64(angle_m * pi) * Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-143)
		tmp = (((angle_m * pi) * -0.011111111111111112) * a) * a;
	else
		tmp = (angle_m * pi) * (((b_m - a) * (a + b_m)) * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-143], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -1.9999999999999999e-143

   1. Initial program 55.3%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 51.8%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot angle, \pi, -\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a}, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right)}{a}\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 51.2

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

   10. Applied rewrites 51.2%

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

       1. lift-*.f64 N/A

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

   12. Applied rewrites 62.5%

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

   if -1.9999999999999999e-143 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 52.2%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.2

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

   4. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 55.3

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

   6. Applied rewrites 55.3%

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

Alternative 13: 57.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -8.15 \cdot 10^{-215}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(b\_m \cdot \left(b\_m - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -8.15e-215)
    (* (* (* (* angle_m PI) -0.011111111111111112) a) a)
    (* (* (* PI angle_m) (* b_m (- b_m a))) 0.011111111111111112))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -8.15e-215) {
		tmp = (((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((((double) M_PI) * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -8.15e-215) {
		tmp = (((angle_m * Math.PI) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((Math.PI * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -8.15e-215:
		tmp = (((angle_m * math.pi) * -0.011111111111111112) * a) * a
	else:
		tmp = ((math.pi * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -8.15e-215)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(b_m * Float64(b_m - a))) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -8.15e-215)
		tmp = (((angle_m * pi) * -0.011111111111111112) * a) * a;
	else
		tmp = ((pi * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -8.15e-215], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b$95$m * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -8.15e-215

   1. Initial program 55.2%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot angle, \pi, -\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a}, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right)}{a}\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 50.9

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

   10. Applied rewrites 50.9%

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

       1. lift-*.f64 N/A

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

   12. Applied rewrites 61.7%

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

   if -8.15e-215 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 52.1%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.5

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in a around 0

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.3%

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

Alternative 14: 57.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -8.15 \cdot 10^{-215}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(b\_m \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -8.15e-215)
    (* (* (* (* angle_m PI) -0.011111111111111112) a) a)
    (* (* (* PI angle_m) (* b_m b_m)) 0.011111111111111112))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -8.15e-215) {
		tmp = (((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((((double) M_PI) * angle_m) * (b_m * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -8.15e-215) {
		tmp = (((angle_m * Math.PI) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((Math.PI * angle_m) * (b_m * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -8.15e-215:
		tmp = (((angle_m * math.pi) * -0.011111111111111112) * a) * a
	else:
		tmp = ((math.pi * angle_m) * (b_m * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -8.15e-215)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(b_m * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -8.15e-215)
		tmp = (((angle_m * pi) * -0.011111111111111112) * a) * a;
	else
		tmp = ((pi * angle_m) * (b_m * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -8.15e-215], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -8.15e-215

   1. Initial program 55.2%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot angle, \pi, -\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a}, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right)}{a}\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 50.9

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

   10. Applied rewrites 50.9%

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

       1. lift-*.f64 N/A

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

   12. Applied rewrites 61.7%

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

   if -8.15e-215 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 52.1%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.5

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in a around 0

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot {b}^{2}\right) \cdot \frac{1}{90} \]
   6. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. pow2 N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.7

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

   7. Applied rewrites 53.7%

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

Alternative 15: 53.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -8.15 \cdot 10^{-215}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -8.15e-215)
    (* (* (* (* angle_m PI) -0.011111111111111112) a) a)
    (* (* (* PI (* b_m b_m)) angle_m) 0.011111111111111112))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -8.15e-215) {
		tmp = (((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((((double) M_PI) * (b_m * b_m)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -8.15e-215) {
		tmp = (((angle_m * Math.PI) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((Math.PI * (b_m * b_m)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -8.15e-215:
		tmp = (((angle_m * math.pi) * -0.011111111111111112) * a) * a
	else:
		tmp = ((math.pi * (b_m * b_m)) * angle_m) * 0.011111111111111112
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -8.15e-215)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a);
	else
		tmp = Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle_m) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -8.15e-215)
		tmp = (((angle_m * pi) * -0.011111111111111112) * a) * a;
	else
		tmp = ((pi * (b_m * b_m)) * angle_m) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -8.15e-215], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -8.15e-215

   1. Initial program 55.2%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot angle, \pi, -\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a}, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right)}{a}\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 50.9

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

   10. Applied rewrites 50.9%

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

       1. lift-*.f64 N/A

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

   12. Applied rewrites 61.7%

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

   if -8.15e-215 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 52.1%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 55.5

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      7. lower-*.f64 53.7

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 53.7%

      \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 38.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (* angle_s (* (* (* (* angle_m PI) -0.011111111111111112) a) a)))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * ((((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a);
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * ((((angle_m * Math.PI) * -0.011111111111111112) * a) * a);
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * ((((angle_m * math.pi) * -0.011111111111111112) * a) * a)

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * ((((angle_m * pi) * -0.011111111111111112) * a) * a);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $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. *-commutative N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

   10. difference-of-squares N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   13. lower--.f64 53.7

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

4. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

5. Taylor expanded in a around -inf

   \[\leadsto {a}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{a} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)}{a} + \frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{\frac{-1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{a} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)}{a} + \frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{\color{blue}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \frac{\frac{-1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{a} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)}{a} + \frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{\color{blue}{2}} \]

7. Applied rewrites 45.0%

   \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot angle, \pi, -\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a}, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right)}{a}\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

8. Taylor expanded in a around inf

   \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

   4. lift-PI.f64 34.8

      \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]

10. Applied rewrites 34.8%

    \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

12. Applied rewrites 38.3%

    \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a \]
13. Add Preprocessing

Alternative 17: 34.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(-0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a \cdot a\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (* angle_s (* (* (* -0.011111111111111112 angle_m) PI) (* a a))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (((-0.011111111111111112 * angle_m) * ((double) M_PI)) * (a * a));
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (((-0.011111111111111112 * angle_m) * Math.PI) * (a * a));
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * (((-0.011111111111111112 * angle_m) * math.pi) * (a * a))

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(-0.011111111111111112 * angle_m) * pi) * Float64(a * a)))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * (((-0.011111111111111112 * angle_m) * pi) * (a * a));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(-0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(a * a), $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. *-commutative N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

   10. difference-of-squares N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   13. lower--.f64 53.7

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

4. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

5. Taylor expanded in a around -inf

   \[\leadsto {a}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{\frac{-1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{a} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)}{a} + \frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{\frac{-1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{a} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)}{a} + \frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{\color{blue}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \frac{\frac{-1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{a} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)}{a} + \frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {a}^{\color{blue}{2}} \]

7. Applied rewrites 45.0%

   \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot angle, \pi, -\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a}, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right)}{a}\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

8. Taylor expanded in a around inf

   \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

   4. lift-PI.f64 34.8

      \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]

10. Applied rewrites 34.8%

    \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]
11. Add Preprocessing

Alternative 18: 13.3% accurate, 17.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \log 1 \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m) :precision binary64 (* angle_s (log 1.0)))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * log(1.0);
}

Fortran

b_m =     private
angle\_m =     private
angle\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(angle_s, a, b_m, angle_m)
use fmin_fmax_functions
    real(8), intent (in) :: angle_s
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle_m
    code = angle_s * log(1.0d0)
end function

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * Math.log(1.0);
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * math.log(1.0)

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * log(1.0))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * log(1.0);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[Log[1.0], $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. *-commutative N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

   10. difference-of-squares N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   13. lower--.f64 53.7

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

4. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

5. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   4. pow2 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   7. lift-PI.f64 34.8

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \pi\right) \]

7. Applied rewrites 34.8%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \pi\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \color{blue}{\pi}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot angle\right) \cdot \mathsf{PI}\left(\right) \]

   5. add-log-exp N/A

      \[\leadsto \left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot angle\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right) \]

   6. log-pow-rev N/A

      \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot angle\right)}\right) \]

   7. lower-log.f64 N/A

      \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot angle\right)}\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot angle\right)}\right) \]

   9. lower-exp.f64 N/A

      \[\leadsto \log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot angle\right)}\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto \log \left({\left(e^{\pi}\right)}^{\left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot angle\right)}\right) \]

   11. lower-*.f64 27.0

       \[\leadsto \log \left({\left(e^{\pi}\right)}^{\left(\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot angle\right)}\right) \]

9. Applied rewrites 27.0%

   \[\leadsto \log \left({\left(e^{\pi}\right)}^{\left(\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot angle\right)}\right) \]

10. Taylor expanded in a around 0

    \[\leadsto \log 1 \]
11. Step-by-step derivation

12. Applied rewrites 13.3%

    \[\leadsto \log 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025113 
(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)))))