ab-angle->ABCF B

Percentage Accurate: 53.7% → 67.3%

Time: 6.2s

Alternatives: 18

Speedup: 5.5×

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.8 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right) \cdot 2\right)\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)\\ \mathbf{elif}\;angle\_m \leq 4.6 \cdot 10^{+285}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\log \left({\left(e^{\pi}\right)}^{0.005555555555555556}\right) \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, 0.5 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 5.8e+29)
    (*
     (*
      (- b a)
      (* (* (+ a b) (sin (* (* 0.005555555555555556 angle_m) PI))) 2.0))
     (cos (* (* PI 0.005555555555555556) angle_m)))
    (if (<= angle_m 4.6e+285)
      (*
       (*
        (* (- b a) (+ b a))
        (* 2.0 (sin (* (* angle_m PI) 0.005555555555555556))))
       (cos (* (log (pow (exp PI) 0.005555555555555556)) angle_m)))
      (*
       (* (* 2.0 (fma (- a) a (* b b))) (sin (* PI (/ angle_m 180.0))))
       (sin (fma (* 0.005555555555555556 angle_m) PI (* 0.5 PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 5.8e+29) {
		tmp = ((b - a) * (((a + b) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)))) * 2.0)) * cos(((((double) M_PI) * 0.005555555555555556) * angle_m));
	} else if (angle_m <= 4.6e+285) {
		tmp = (((b - a) * (b + a)) * (2.0 * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)))) * cos((log(pow(exp(((double) M_PI)), 0.005555555555555556)) * angle_m));
	} else {
		tmp = ((2.0 * fma(-a, a, (b * b))) * sin((((double) M_PI) * (angle_m / 180.0)))) * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (0.5 * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 5.8e+29)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))) * 2.0)) * cos(Float64(Float64(pi * 0.005555555555555556) * angle_m)));
	elseif (angle_m <= 4.6e+285)
		tmp = Float64(Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(2.0 * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)))) * cos(Float64(log((exp(pi) ^ 0.005555555555555556)) * angle_m)));
	else
		tmp = Float64(Float64(Float64(2.0 * fma(Float64(-a), a, Float64(b * b))) * sin(Float64(pi * Float64(angle_m / 180.0)))) * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(0.5 * pi))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 5.8e+29], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 4.6e+285], N[(N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[Log[N[Power[N[Exp[Pi], $MachinePrecision], 0.005555555555555556], $MachinePrecision]], $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[((-a) * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 5.7999999999999999e29

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 67.3%

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

   if 5.7999999999999999e29 < angle < 4.6000000000000001e285

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-log-exp N/A

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

      5. log-pow-rev N/A

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

      6. lower-log.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-exp.f64 57.3

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

   9. Applied rewrites 57.3%

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

   if 4.6000000000000001e285 < angle

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 56.4

          \[\leadsto \left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \pi \cdot 0.5\right)\right)} \]

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 56.4

          \[\leadsto \left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \color{blue}{0.5 \cdot \pi}\right)\right) \]

   5. Applied rewrites 56.4%

      \[\leadsto \left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, 0.5 \cdot \pi\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right) \cdot 2\right)\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+263}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 2e+223)
    (*
     (*
      (- b a)
      (* (* (+ a b) (sin (* (* 0.005555555555555556 angle_m) PI))) 2.0))
     (cos (* (* PI 0.005555555555555556) angle_m)))
    (if (<= a 4.5e+263)
      (* 0.011111111111111112 (* (+ b a) (* (* (- b a) PI) angle_m)))
      (*
       (*
        (* (- b a) (+ b a))
        (* 2.0 (sin (* (* angle_m PI) 0.005555555555555556))))
       1.0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2e+223) {
		tmp = ((b - a) * (((a + b) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)))) * 2.0)) * cos(((((double) M_PI) * 0.005555555555555556) * angle_m));
	} else if (a <= 4.5e+263) {
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * ((double) M_PI)) * angle_m));
	} else {
		tmp = (((b - a) * (b + a)) * (2.0 * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)))) * 1.0;
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 2e+223) {
		tmp = ((b - a) * (((a + b) * Math.sin(((0.005555555555555556 * angle_m) * Math.PI))) * 2.0)) * Math.cos(((Math.PI * 0.005555555555555556) * angle_m));
	} else if (a <= 4.5e+263) {
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * Math.PI) * angle_m));
	} else {
		tmp = (((b - a) * (b + a)) * (2.0 * Math.sin(((angle_m * Math.PI) * 0.005555555555555556)))) * 1.0;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 2e+223:
		tmp = ((b - a) * (((a + b) * math.sin(((0.005555555555555556 * angle_m) * math.pi))) * 2.0)) * math.cos(((math.pi * 0.005555555555555556) * angle_m))
	elif a <= 4.5e+263:
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * math.pi) * angle_m))
	else:
		tmp = (((b - a) * (b + a)) * (2.0 * math.sin(((angle_m * math.pi) * 0.005555555555555556)))) * 1.0
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 2e+223)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))) * 2.0)) * cos(Float64(Float64(pi * 0.005555555555555556) * angle_m)));
	elseif (a <= 4.5e+263)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(Float64(Float64(b - a) * pi) * angle_m)));
	else
		tmp = Float64(Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(2.0 * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)))) * 1.0);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 2e+223)
		tmp = ((b - a) * (((a + b) * sin(((0.005555555555555556 * angle_m) * pi))) * 2.0)) * cos(((pi * 0.005555555555555556) * angle_m));
	elseif (a <= 4.5e+263)
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * pi) * angle_m));
	else
		tmp = (((b - a) * (b + a)) * (2.0 * sin(((angle_m * pi) * 0.005555555555555556)))) * 1.0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 2e+223], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.5e+263], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < 2.00000000000000009e223

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 67.3%

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

   if 2.00000000000000009e223 < a < 4.50000000000000014e263

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 4.50000000000000014e263 < a

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 55.4%

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

Alternative 3: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right)\right) \cdot \pi\\ \mathbf{elif}\;angle\_m \leq 1.55 \cdot 10^{+147}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \left(2 \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\frac{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot b}{a + b} \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2.6e-37)
    (* (* (- b a) (* (* (+ a b) 0.011111111111111112) angle_m)) PI)
    (if (<= angle_m 1.55e+147)
      (*
       (*
        (* (- b a) (+ b a))
        (* 2.0 (sin (* (* angle_m PI) 0.005555555555555556))))
       (cos (* (* PI 0.005555555555555556) angle_m)))
      (*
       0.011111111111111112
       (* (* (/ (* (* (- b a) (+ a b)) b) (+ a b)) PI) angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.6e-37) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * ((double) M_PI);
	} else if (angle_m <= 1.55e+147) {
		tmp = (((b - a) * (b + a)) * (2.0 * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)))) * cos(((((double) M_PI) * 0.005555555555555556) * angle_m));
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.6e-37) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * Math.PI;
	} else if (angle_m <= 1.55e+147) {
		tmp = (((b - a) * (b + a)) * (2.0 * Math.sin(((angle_m * Math.PI) * 0.005555555555555556)))) * Math.cos(((Math.PI * 0.005555555555555556) * angle_m));
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 2.6e-37:
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * math.pi
	elif angle_m <= 1.55e+147:
		tmp = (((b - a) * (b + a)) * (2.0 * math.sin(((angle_m * math.pi) * 0.005555555555555556)))) * math.cos(((math.pi * 0.005555555555555556) * angle_m))
	else:
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 2.6e-37)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * 0.011111111111111112) * angle_m)) * pi);
	elseif (angle_m <= 1.55e+147)
		tmp = Float64(Float64(Float64(Float64(b - a) * Float64(b + a)) * Float64(2.0 * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)))) * cos(Float64(Float64(pi * 0.005555555555555556) * angle_m)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(Float64(Float64(b - a) * Float64(a + b)) * b) / Float64(a + b)) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 2.6e-37)
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * pi;
	elseif (angle_m <= 1.55e+147)
		tmp = (((b - a) * (b + a)) * (2.0 * sin(((angle_m * pi) * 0.005555555555555556)))) * cos(((pi * 0.005555555555555556) * angle_m));
	else
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2.6e-37], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], If[LessEqual[angle$95$m, 1.55e+147], N[(N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 2.5999999999999998e-37

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 62.3

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

   10. Applied rewrites 62.3%

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

   if 2.5999999999999998e-37 < angle < 1.55e147

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

   if 1.55e147 < angle

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. flip-- N/A

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

       5. unpow2 N/A

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. pow-to-exp N/A

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

       9. lift-log.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift--.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. associate-*l/ N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 38.4%

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

Alternative 4: 66.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 6 \cdot 10^{+169}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right) \cdot 2\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin \left(\frac{1}{\frac{180}{angle\_m \cdot \pi}}\right)\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (cos (* (* PI 0.005555555555555556) angle_m))))
   (*
    angle_s
    (if (<= angle_m 6e+169)
      (*
       (*
        (- b a)
        (* (* (+ a b) (sin (* (* 0.005555555555555556 angle_m) PI))) 2.0))
       t_0)
      (*
       (* (* 2.0 (fma (- a) a (* b b))) (sin (/ 1.0 (/ 180.0 (* angle_m PI)))))
       t_0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = cos(((((double) M_PI) * 0.005555555555555556) * angle_m));
	double tmp;
	if (angle_m <= 6e+169) {
		tmp = ((b - a) * (((a + b) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)))) * 2.0)) * t_0;
	} else {
		tmp = ((2.0 * fma(-a, a, (b * b))) * sin((1.0 / (180.0 / (angle_m * ((double) M_PI)))))) * t_0;
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = cos(Float64(Float64(pi * 0.005555555555555556) * angle_m))
	tmp = 0.0
	if (angle_m <= 6e+169)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))) * 2.0)) * t_0);
	else
		tmp = Float64(Float64(Float64(2.0 * fma(Float64(-a), a, Float64(b * b))) * sin(Float64(1.0 / Float64(180.0 / Float64(angle_m * pi))))) * t_0);
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Cos[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 6e+169], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(2.0 * N[((-a) * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(1.0 / N[(180.0 / N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 5.9999999999999999e169

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 67.3%

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

   if 5.9999999999999999e169 < angle

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-flip N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 56.2

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

   7. Applied rewrites 56.2%

      \[\leadsto \left(\left(2 \cdot \mathsf{fma}\left(-a, a, b \cdot b\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)}\right) \cdot \cos \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.2 \cdot 10^{-68}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right)\right) \cdot \pi\\ \mathbf{elif}\;angle\_m \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;\left(\cos \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\frac{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot b}{a + b} \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 3.2e-68)
    (* (* (- b a) (* (* (+ a b) 0.011111111111111112) angle_m)) PI)
    (if (<= angle_m 4.3e+139)
      (*
       (*
        (cos (* -0.005555555555555556 (* angle_m PI)))
        (sin (* (* PI 0.005555555555555556) angle_m)))
       (* (* (+ b a) (- b a)) 2.0))
      (*
       0.011111111111111112
       (* (* (/ (* (* (- b a) (+ a b)) b) (+ a b)) PI) angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.2e-68) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * ((double) M_PI);
	} else if (angle_m <= 4.3e+139) {
		tmp = (cos((-0.005555555555555556 * (angle_m * ((double) M_PI)))) * sin(((((double) M_PI) * 0.005555555555555556) * angle_m))) * (((b + a) * (b - a)) * 2.0);
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.2e-68) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * Math.PI;
	} else if (angle_m <= 4.3e+139) {
		tmp = (Math.cos((-0.005555555555555556 * (angle_m * Math.PI))) * Math.sin(((Math.PI * 0.005555555555555556) * angle_m))) * (((b + a) * (b - a)) * 2.0);
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 3.2e-68:
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * math.pi
	elif angle_m <= 4.3e+139:
		tmp = (math.cos((-0.005555555555555556 * (angle_m * math.pi))) * math.sin(((math.pi * 0.005555555555555556) * angle_m))) * (((b + a) * (b - a)) * 2.0)
	else:
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 3.2e-68)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * 0.011111111111111112) * angle_m)) * pi);
	elseif (angle_m <= 4.3e+139)
		tmp = Float64(Float64(cos(Float64(-0.005555555555555556 * Float64(angle_m * pi))) * sin(Float64(Float64(pi * 0.005555555555555556) * angle_m))) * Float64(Float64(Float64(b + a) * Float64(b - a)) * 2.0));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(Float64(Float64(b - a) * Float64(a + b)) * b) / Float64(a + b)) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 3.2e-68)
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * pi;
	elseif (angle_m <= 4.3e+139)
		tmp = (cos((-0.005555555555555556 * (angle_m * pi))) * sin(((pi * 0.005555555555555556) * angle_m))) * (((b + a) * (b - a)) * 2.0);
	else
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 3.2e-68], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], If[LessEqual[angle$95$m, 4.3e+139], N[(N[(N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 3.1999999999999999e-68

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 62.3

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

   10. Applied rewrites 62.3%

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

   if 3.1999999999999999e-68 < angle < 4.2999999999999998e139

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

   5. Applied rewrites 57.5%

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

   if 4.2999999999999998e139 < angle

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. flip-- N/A

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

       5. unpow2 N/A

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. pow-to-exp N/A

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

       9. lift-log.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift--.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. associate-*l/ N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 38.4%

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

Alternative 6: 66.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right)\right) \cdot \pi\\ \mathbf{elif}\;angle\_m \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;\left(\cos \left(-0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right) \cdot \left(\left(2 \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\frac{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot b}{a + b} \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2.6e-37)
    (* (* (- b a) (* (* (+ a b) 0.011111111111111112) angle_m)) PI)
    (if (<= angle_m 4.3e+139)
      (*
       (*
        (cos (* -0.005555555555555556 (* angle_m PI)))
        (sin (* (* angle_m PI) 0.005555555555555556)))
       (* (* 2.0 (- b a)) (+ b a)))
      (*
       0.011111111111111112
       (* (* (/ (* (* (- b a) (+ a b)) b) (+ a b)) PI) angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.6e-37) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * ((double) M_PI);
	} else if (angle_m <= 4.3e+139) {
		tmp = (cos((-0.005555555555555556 * (angle_m * ((double) M_PI)))) * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556))) * ((2.0 * (b - a)) * (b + a));
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.6e-37) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * Math.PI;
	} else if (angle_m <= 4.3e+139) {
		tmp = (Math.cos((-0.005555555555555556 * (angle_m * Math.PI))) * Math.sin(((angle_m * Math.PI) * 0.005555555555555556))) * ((2.0 * (b - a)) * (b + a));
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 2.6e-37:
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * math.pi
	elif angle_m <= 4.3e+139:
		tmp = (math.cos((-0.005555555555555556 * (angle_m * math.pi))) * math.sin(((angle_m * math.pi) * 0.005555555555555556))) * ((2.0 * (b - a)) * (b + a))
	else:
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 2.6e-37)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * 0.011111111111111112) * angle_m)) * pi);
	elseif (angle_m <= 4.3e+139)
		tmp = Float64(Float64(cos(Float64(-0.005555555555555556 * Float64(angle_m * pi))) * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556))) * Float64(Float64(2.0 * Float64(b - a)) * Float64(b + a)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(Float64(Float64(b - a) * Float64(a + b)) * b) / Float64(a + b)) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 2.6e-37)
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * pi;
	elseif (angle_m <= 4.3e+139)
		tmp = (cos((-0.005555555555555556 * (angle_m * pi))) * sin(((angle_m * pi) * 0.005555555555555556))) * ((2.0 * (b - a)) * (b + a));
	else
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2.6e-37], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], If[LessEqual[angle$95$m, 4.3e+139], N[(N[(N[Cos[N[(-0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 2.5999999999999998e-37

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 62.3

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

   10. Applied rewrites 62.3%

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

   if 2.5999999999999998e-37 < angle < 4.2999999999999998e139

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

   7. Applied rewrites 57.3%

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

   if 4.2999999999999998e139 < angle

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. flip-- N/A

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

       5. unpow2 N/A

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. pow-to-exp N/A

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

       9. lift-log.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift--.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. associate-*l/ N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 38.4%

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

Alternative 7: 65.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.2 \cdot 10^{-68}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right)\right) \cdot \pi\\ \mathbf{elif}\;angle\_m \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\frac{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot b}{a + b} \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 3.2e-68)
    (* (* (- b a) (* (* (+ a b) 0.011111111111111112) angle_m)) PI)
    (if (<= angle_m 1.8e+147)
      (*
       (* (+ b a) (- b a))
       (sin (* 2.0 (* (* PI 0.005555555555555556) angle_m))))
      (*
       0.011111111111111112
       (* (* (/ (* (* (- b a) (+ a b)) b) (+ a b)) PI) angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.2e-68) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * ((double) M_PI);
	} else if (angle_m <= 1.8e+147) {
		tmp = ((b + a) * (b - a)) * sin((2.0 * ((((double) M_PI) * 0.005555555555555556) * angle_m)));
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.2e-68) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * Math.PI;
	} else if (angle_m <= 1.8e+147) {
		tmp = ((b + a) * (b - a)) * Math.sin((2.0 * ((Math.PI * 0.005555555555555556) * angle_m)));
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 3.2e-68:
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * math.pi
	elif angle_m <= 1.8e+147:
		tmp = ((b + a) * (b - a)) * math.sin((2.0 * ((math.pi * 0.005555555555555556) * angle_m)))
	else:
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 3.2e-68)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * 0.011111111111111112) * angle_m)) * pi);
	elseif (angle_m <= 1.8e+147)
		tmp = Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(2.0 * Float64(Float64(pi * 0.005555555555555556) * angle_m))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(Float64(Float64(b - a) * Float64(a + b)) * b) / Float64(a + b)) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 3.2e-68)
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * pi;
	elseif (angle_m <= 1.8e+147)
		tmp = ((b + a) * (b - a)) * sin((2.0 * ((pi * 0.005555555555555556) * angle_m)));
	else
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 3.2e-68], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], If[LessEqual[angle$95$m, 1.8e+147], N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 3.1999999999999999e-68

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 62.3

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

   10. Applied rewrites 62.3%

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

   if 3.1999999999999999e-68 < angle < 1.8000000000000001e147

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

   5. Applied rewrites 57.5%

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

   if 1.8000000000000001e147 < angle

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. flip-- N/A

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

       5. unpow2 N/A

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. pow-to-exp N/A

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

       9. lift-log.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift--.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. associate-*l/ N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 38.4%

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

Alternative 8: 65.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;\left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\left(\left(a + b\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\frac{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot b}{a + b} \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.3e+139)
    (*
     (* (cos (* -0.005555555555555556 (* PI angle_m))) (- b a))
     (* (* (+ a b) (sin (* (* 0.005555555555555556 angle_m) PI))) 2.0))
    (*
     0.011111111111111112
     (* (* (/ (* (* (- b a) (+ a b)) b) (+ a b)) PI) angle_m)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.3e+139) {
		tmp = (cos((-0.005555555555555556 * (((double) M_PI) * angle_m))) * (b - a)) * (((a + b) * sin(((0.005555555555555556 * angle_m) * ((double) M_PI)))) * 2.0);
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.3e+139) {
		tmp = (Math.cos((-0.005555555555555556 * (Math.PI * angle_m))) * (b - a)) * (((a + b) * Math.sin(((0.005555555555555556 * angle_m) * Math.PI))) * 2.0);
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 4.3e+139:
		tmp = (math.cos((-0.005555555555555556 * (math.pi * angle_m))) * (b - a)) * (((a + b) * math.sin(((0.005555555555555556 * angle_m) * math.pi))) * 2.0)
	else:
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.3e+139)
		tmp = Float64(Float64(cos(Float64(-0.005555555555555556 * Float64(pi * angle_m))) * Float64(b - a)) * Float64(Float64(Float64(a + b) * sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))) * 2.0));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(Float64(Float64(b - a) * Float64(a + b)) * b) / Float64(a + b)) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 4.3e+139)
		tmp = (cos((-0.005555555555555556 * (pi * angle_m))) * (b - a)) * (((a + b) * sin(((0.005555555555555556 * angle_m) * pi))) * 2.0);
	else
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.3e+139], N[(N[(N[Cos[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.2999999999999998e139

   1. Initial program 53.7%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 56.4

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 56.4

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

   3. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 56.1

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

   5. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 57.4%

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

   9. Applied rewrites 67.5%

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

   if 4.2999999999999998e139 < angle

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.3

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. flip-- N/A

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

       5. unpow2 N/A

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

       6. lift-pow.f64 N/A

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

       7. unpow2 N/A

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

       8. pow-to-exp N/A

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

       9. lift-log.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-exp.f64 N/A

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

       12. lift--.f64 N/A

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

       13. lift-+.f64 N/A

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

       14. associate-*l/ N/A

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

       15. lower-/.f64 N/A

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

   11. Applied rewrites 38.4%

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

Alternative 9: 63.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.85 \cdot 10^{+81}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right)\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\frac{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot b}{a + b} \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.85e+81)
    (* (* (- b a) (* (* (+ a b) 0.011111111111111112) angle_m)) PI)
    (*
     0.011111111111111112
     (* (* (/ (* (* (- b a) (+ a b)) b) (+ a b)) PI) angle_m)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.85e+81) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * ((double) M_PI);
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.85e+81) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * Math.PI;
	} else {
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 1.85e+81:
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * math.pi
	else:
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.85e+81)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * 0.011111111111111112) * angle_m)) * pi);
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(Float64(Float64(b - a) * Float64(a + b)) * b) / Float64(a + b)) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.85e+81)
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * pi;
	else
		tmp = 0.011111111111111112 * ((((((b - a) * (a + b)) * b) / (a + b)) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.85e+81], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.85e81

   1. Initial program 53.7%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      6. lower-*.f64 62.3

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      8. *-commutative N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)\right) \cdot \pi \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)\right) \cdot \pi \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot 0.011111111111111112\right) \cdot angle\right)\right) \cdot \pi \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      14. +-commutative N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      15. lower-+.f64 62.3

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\right)\right) \cdot \pi \]

   10. Applied rewrites 62.3%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\right)\right) \cdot \color{blue}{\pi} \]

   if 1.85e81 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.9%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       3. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b - a\right) \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       4. flip-- N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{b \cdot b - a \cdot a}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - a \cdot a}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - a \cdot a}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - {a}^{2}}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       8. pow-to-exp N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - e^{\log a \cdot 2}}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       9. lift-log.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - e^{\log a \cdot 2}}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - e^{\log a \cdot 2}}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       11. lift-exp.f64 N/A

           \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - e^{\log a \cdot 2}}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       12. lift--.f64 N/A

           \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - e^{\log a \cdot 2}}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       13. lift-+.f64 N/A

           \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\frac{{b}^{2} - e^{\log a \cdot 2}}{b + a} \cdot b\right) \cdot \pi\right) \cdot angle\right) \]

       14. associate-*l/ N/A

           \[\leadsto \frac{1}{90} \cdot \left(\left(\frac{\left({b}^{2} - e^{\log a \cdot 2}\right) \cdot b}{b + a} \cdot \pi\right) \cdot angle\right) \]

       15. lower-/.f64 N/A

           \[\leadsto \frac{1}{90} \cdot \left(\left(\frac{\left({b}^{2} - e^{\log a \cdot 2}\right) \cdot b}{b + a} \cdot \pi\right) \cdot angle\right) \]

   11. Applied rewrites 38.4%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(\frac{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot b}{a + b} \cdot \pi\right) \cdot angle\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right)\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2e-37)
    (* (* (- b a) (* (* (+ a b) 0.011111111111111112) angle_m)) PI)
    (* (* (* angle_m (* (+ b a) (- b a))) PI) 0.011111111111111112))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2e-37) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * ((double) M_PI);
	} else {
		tmp = ((angle_m * ((b + a) * (b - a))) * ((double) M_PI)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2e-37) {
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * Math.PI;
	} else {
		tmp = ((angle_m * ((b + a) * (b - a))) * Math.PI) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 2e-37:
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * math.pi
	else:
		tmp = ((angle_m * ((b + a) * (b - a))) * math.pi) * 0.011111111111111112
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 2e-37)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(Float64(a + b) * 0.011111111111111112) * angle_m)) * pi);
	else
		tmp = Float64(Float64(Float64(angle_m * Float64(Float64(b + a) * Float64(b - a))) * pi) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 2e-37)
		tmp = ((b - a) * (((a + b) * 0.011111111111111112) * angle_m)) * pi;
	else
		tmp = ((angle_m * ((b + a) * (b - a))) * pi) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2e-37], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(N[(angle$95$m * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.00000000000000013e-37

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      6. lower-*.f64 62.3

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right)\right) \cdot \pi \]

      8. *-commutative N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)\right) \cdot \pi \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)\right) \cdot \pi \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot 0.011111111111111112\right) \cdot angle\right)\right) \cdot \pi \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      14. +-commutative N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot \frac{1}{90}\right) \cdot angle\right)\right) \cdot \pi \]

      15. lower-+.f64 62.3

          \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\right)\right) \cdot \pi \]

   10. Applied rewrites 62.3%

       \[\leadsto \left(\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 0.011111111111111112\right) \cdot angle\right)\right) \cdot \color{blue}{\pi} \]

   if 2.00000000000000013e-37 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. lower-*.f64 50.3

         \[\leadsto \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{0.011111111111111112} \]

   6. Applied rewrites 54.2%

      \[\leadsto \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \cdot \color{blue}{0.011111111111111112} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.5 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.5e-69)
    (* (* (* 0.011111111111111112 angle_m) (+ b a)) (* (- b a) PI))
    (* (* (* 0.011111111111111112 angle_m) PI) (* (+ b a) (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.5e-69) {
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * ((b - a) * ((double) M_PI));
	} else {
		tmp = ((0.011111111111111112 * angle_m) * ((double) M_PI)) * ((b + a) * (b - a));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.5e-69) {
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * ((b - a) * Math.PI);
	} else {
		tmp = ((0.011111111111111112 * angle_m) * Math.PI) * ((b + a) * (b - a));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 4.5e-69:
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * ((b - a) * math.pi)
	else:
		tmp = ((0.011111111111111112 * angle_m) * math.pi) * ((b + a) * (b - a))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.5e-69)
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(b + a)) * Float64(Float64(b - a) * pi));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * pi) * Float64(Float64(b + a) * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 4.5e-69)
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * ((b - a) * pi);
	else
		tmp = ((0.011111111111111112 * angle_m) * pi) * ((b + a) * (b - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.5e-69], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.50000000000000009e-69

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   if 4.50000000000000009e-69 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right) \]

      6. pow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{{a}^{2}}\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {a}^{\color{blue}{2}}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      11. fp-cancel-sub-sign N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot a}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot a\right)\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \left(-a\right) \cdot a\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \left(-a\right) \cdot \color{blue}{a}\right)\right) \]

      15. lift-fma.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, \color{blue}{b}, \left(-a\right) \cdot a\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\color{blue}{b}, b, \left(-a\right) \cdot a\right) \]

      19. lower-*.f64 52.7

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right) \]

      20. lift-fma.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b + \color{blue}{\left(-a\right) \cdot a}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.6 \cdot 10^{-69}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.6e-69)
    (* 0.011111111111111112 (* (+ b a) (* (* (- b a) PI) angle_m)))
    (* (* (* 0.011111111111111112 angle_m) PI) (* (+ b a) (- b a))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e-69) {
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * ((double) M_PI)) * angle_m));
	} else {
		tmp = ((0.011111111111111112 * angle_m) * ((double) M_PI)) * ((b + a) * (b - a));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e-69) {
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * Math.PI) * angle_m));
	} else {
		tmp = ((0.011111111111111112 * angle_m) * Math.PI) * ((b + a) * (b - a));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 4.6e-69:
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * math.pi) * angle_m))
	else:
		tmp = ((0.011111111111111112 * angle_m) * math.pi) * ((b + a) * (b - a))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.6e-69)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(Float64(Float64(b - a) * pi) * angle_m)));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * pi) * Float64(Float64(b + a) * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 4.6e-69)
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * pi) * angle_m));
	else
		tmp = ((0.011111111111111112 * angle_m) * pi) * ((b + a) * (b - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.6e-69], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.6000000000000001e-69

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot angle\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot \color{blue}{angle}\right)\right) \]

      8. lower-*.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)\right) \]

   8. Applied rewrites 62.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

   if 4.6000000000000001e-69 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right) \]

      6. pow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - \color{blue}{{a}^{2}}\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {a}^{\color{blue}{2}}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      11. fp-cancel-sub-sign N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot a}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot a\right)\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \left(-a\right) \cdot a\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b + \left(-a\right) \cdot \color{blue}{a}\right)\right) \]

      15. lift-fma.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, \color{blue}{b}, \left(-a\right) \cdot a\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\color{blue}{b}, b, \left(-a\right) \cdot a\right) \]

      19. lower-*.f64 52.7

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \mathsf{fma}\left(b, b, \left(-a\right) \cdot a\right) \]

      20. lift-fma.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \pi\right) \cdot \left(b \cdot b + \color{blue}{\left(-a\right) \cdot a}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.6 \cdot 10^{-69}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.6e-69)
    (* 0.011111111111111112 (* (+ b a) (* (* (- b a) PI) angle_m)))
    (* 0.011111111111111112 (* (* (* (+ b a) (- b a)) PI) angle_m)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e-69) {
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * ((double) M_PI)) * angle_m));
	} else {
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e-69) {
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * Math.PI) * angle_m));
	} else {
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 4.6e-69:
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * math.pi) * angle_m))
	else:
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.6e-69)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(Float64(Float64(b - a) * pi) * angle_m)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 4.6e-69)
		tmp = 0.011111111111111112 * ((b + a) * (((b - a) * pi) * angle_m));
	else
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.6e-69], N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.6000000000000001e-69

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot angle\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot \color{blue}{angle}\right)\right) \]

      8. lower-*.f64 62.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)\right) \]

   8. Applied rewrites 62.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

   if 4.6000000000000001e-69 < angle

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(a \cdot angle\_m\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+148}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \pi\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
   (*
    angle_s
    (if (<= t_0 (- INFINITY))
      (* (* 0.011111111111111112 (* a angle_m)) (* (- b a) PI))
      (if (<= t_0 2e+148)
        (* 0.011111111111111112 (* (* (* (+ b a) (- b a)) PI) angle_m))
        (* (* (* 0.011111111111111112 angle_m) (+ b a)) (* b PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * ((double) M_PI));
	} else if (t_0 <= 2e+148) {
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * ((double) M_PI)) * angle_m);
	} else {
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * ((double) M_PI));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * Math.PI);
	} else if (t_0 <= 2e+148) {
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * Math.PI) * angle_m);
	} else {
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * Math.PI);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * math.pi)
	elif t_0 <= 2e+148:
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * math.pi) * angle_m)
	else:
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * math.pi)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.011111111111111112 * Float64(a * angle_m)) * Float64(Float64(b - a) * pi));
	elseif (t_0 <= 2e+148)
		tmp = Float64(0.011111111111111112 * Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * pi) * angle_m));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(b + a)) * Float64(b * pi));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * pi);
	elseif (t_0 <= 2e+148)
		tmp = 0.011111111111111112 * ((((b + a) * (b - a)) * pi) * angle_m);
	else
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * pi);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.011111111111111112 * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+148], N[(0.011111111111111112 * N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $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)))) < -inf.0

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - \color{blue}{a}\right) \cdot \pi\right) \]

       2. lower-*.f64 39.3

          \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right) \]

   11. Applied rewrites 39.3%

       \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.0000000000000001e148

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

   if 2.0000000000000001e148 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) \]

       2. lower-PI.f64 42.1

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \pi\right) \]

   11. Applied rewrites 42.1%

       \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \color{blue}{\pi}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-288}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(a \cdot angle\_m\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-288)
    (* (* 0.011111111111111112 (* a angle_m)) (* (- b a) PI))
    (* (* (* 0.011111111111111112 angle_m) (+ b a)) (* b PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-288) {
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * ((double) M_PI));
	} else {
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * ((double) M_PI));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 1e-288) {
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * Math.PI);
	} else {
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * Math.PI);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 1e-288:
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * math.pi)
	else:
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * math.pi)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-288)
		tmp = Float64(Float64(0.011111111111111112 * Float64(a * angle_m)) * Float64(Float64(b - a) * pi));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * angle_m) * Float64(b + a)) * Float64(b * pi));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 1e-288)
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * pi);
	else
		tmp = ((0.011111111111111112 * angle_m) * (b + a)) * (b * pi);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-288], N[(N[(0.011111111111111112 * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b * Pi), $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)))) < 1.00000000000000006e-288

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - \color{blue}{a}\right) \cdot \pi\right) \]

       2. lower-*.f64 39.3

          \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right) \]

   11. Applied rewrites 39.3%

       \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

   if 1.00000000000000006e-288 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) \]

       2. lower-PI.f64 42.1

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \pi\right) \]

   11. Applied rewrites 42.1%

       \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(b \cdot \color{blue}{\pi}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 60.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot \pi\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-288}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(a \cdot angle\_m\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(angle\_m \cdot b\right)\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (- b a) PI)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-288)
      (* (* 0.011111111111111112 (* a angle_m)) t_0)
      (* (* 0.011111111111111112 (* angle_m b)) t_0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b - a) * ((double) M_PI);
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-288) {
		tmp = (0.011111111111111112 * (a * angle_m)) * t_0;
	} else {
		tmp = (0.011111111111111112 * (angle_m * b)) * t_0;
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b - a) * Math.PI;
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 1e-288) {
		tmp = (0.011111111111111112 * (a * angle_m)) * t_0;
	} else {
		tmp = (0.011111111111111112 * (angle_m * b)) * t_0;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b - a) * math.pi
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 1e-288:
		tmp = (0.011111111111111112 * (a * angle_m)) * t_0
	else:
		tmp = (0.011111111111111112 * (angle_m * b)) * t_0
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b - a) * pi)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-288)
		tmp = Float64(Float64(0.011111111111111112 * Float64(a * angle_m)) * t_0);
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(angle_m * b)) * t_0);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (b - a) * pi;
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 1e-288)
		tmp = (0.011111111111111112 * (a * angle_m)) * t_0;
	else
		tmp = (0.011111111111111112 * (angle_m * b)) * t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-288], N[(N[(0.011111111111111112 * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(0.011111111111111112 * N[(angle$95$m * b), $MachinePrecision]), $MachinePrecision] * t$95$0), $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)))) < 1.00000000000000006e-288

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - \color{blue}{a}\right) \cdot \pi\right) \]

       2. lower-*.f64 39.3

          \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right) \]

   11. Applied rewrites 39.3%

       \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

   if 1.00000000000000006e-288 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot b\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot b\right)\right) \cdot \left(\left(b - \color{blue}{a}\right) \cdot \pi\right) \]

       2. lower-*.f64 40.5

          \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot b\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right) \]

   11. Applied rewrites 40.5%

       \[\leadsto \left(0.011111111111111112 \cdot \left(angle \cdot b\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 60.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-288}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(a \cdot angle\_m\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-288)
    (* (* 0.011111111111111112 (* a angle_m)) (* (- b a) PI))
    (* 0.011111111111111112 (* b (* (* PI (- b a)) angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-288) {
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * ((double) M_PI));
	} else {
		tmp = 0.011111111111111112 * (b * ((((double) M_PI) * (b - a)) * angle_m));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 1e-288) {
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * Math.PI);
	} else {
		tmp = 0.011111111111111112 * (b * ((Math.PI * (b - a)) * angle_m));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 1e-288:
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * math.pi)
	else:
		tmp = 0.011111111111111112 * (b * ((math.pi * (b - a)) * angle_m))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-288)
		tmp = Float64(Float64(0.011111111111111112 * Float64(a * angle_m)) * Float64(Float64(b - a) * pi));
	else
		tmp = Float64(0.011111111111111112 * Float64(b * Float64(Float64(pi * Float64(b - a)) * angle_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 1e-288)
		tmp = (0.011111111111111112 * (a * angle_m)) * ((b - a) * pi);
	else
		tmp = 0.011111111111111112 * (b * ((pi * (b - a)) * angle_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-288], N[(N[(0.011111111111111112 * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(b * N[(N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]), $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)))) < 1.00000000000000006e-288

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(\color{blue}{b} - a\right) \cdot \pi\right) \]

      12. lower-*.f64 62.3

          \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\pi}\right) \]

   8. Applied rewrites 62.3%

      \[\leadsto \left(\left(0.011111111111111112 \cdot angle\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \pi\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - \color{blue}{a}\right) \cdot \pi\right) \]

       2. lower-*.f64 39.3

          \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \pi\right) \]

   11. Applied rewrites 39.3%

       \[\leadsto \left(0.011111111111111112 \cdot \left(a \cdot angle\right)\right) \cdot \left(\color{blue}{\left(b - a\right)} \cdot \pi\right) \]

   if 1.00000000000000006e-288 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

      3. lower-*.f64 50.3

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.9%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

       4. associate-*l* N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot angle\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

       8. lower-*.f64 41.6

          \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot \color{blue}{angle}\right)\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(b \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{90} \cdot \left(b \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\right)\right) \]

       11. lower-*.f64 41.6

           \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\right)\right) \]

   11. Applied rewrites 41.6%

       \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \color{blue}{\left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 41.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\_m\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* b (* (* PI (- b a)) angle_m)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (b * ((((double) M_PI) * (b - a)) * angle_m)));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (b * ((Math.PI * (b - a)) * angle_m)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (b * ((math.pi * (b - a)) * angle_m)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(b * Float64(Float64(pi * Float64(b - a)) * angle_m))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (b * ((pi * (b - a)) * angle_m)));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(b * N[(N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

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. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

   3. lower-*.f64 50.3

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{angle}\right) \]

6. Applied rewrites 54.2%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]
8. Step-by-step derivation

9. Applied rewrites 37.9%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot \color{blue}{angle}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b \cdot \left(b - a\right)\right) \cdot \pi\right) \cdot angle\right) \]

    4. associate-*l* N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot angle\right) \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(b \cdot \left(\left(b - a\right) \cdot \pi\right)\right) \cdot angle\right) \]

    6. associate-*l* N/A

       \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)}\right) \]

    8. lower-*.f64 41.6

       \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot \color{blue}{angle}\right)\right) \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(b \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot angle\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \frac{1}{90} \cdot \left(b \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\right)\right) \]

    11. lower-*.f64 41.6

        \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\right)\right) \]

11. Applied rewrites 41.6%

    \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \color{blue}{\left(\left(\pi \cdot \left(b - a\right)\right) \cdot angle\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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)))))