ab-angle->ABCF B

Percentage Accurate: 53.3% → 67.1%

Time: 7.4s

Alternatives: 16

Speedup: 5.5×

• 32.0% 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: 53.3% 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: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(a + b\_m\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\\ t_1 := \left(\left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right) \cdot 2\right) \cdot t\_0\right) \cdot \left(b\_m - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 1.35 \cdot 10^{-214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b\_m \leq 5.5 \cdot 10^{+196}:\\ \;\;\;\;\left(\left(\sin \left(\left(-\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) + \frac{\pi}{2}\right) \cdot 2\right) \cdot t\_0\right) \cdot \left(b\_m - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (+ a b_m) (sin (* (* angle_m PI) 0.005555555555555556))))
        (t_1
         (*
          (*
           (* (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))) 2.0)
           t_0)
          (- b_m a))))
   (*
    angle_s
    (if (<= b_m 1.35e-214)
      t_1
      (if (<= b_m 5.5e+196)
        (*
         (*
          (*
           (sin (+ (- (* (* 0.005555555555555556 angle_m) PI)) (/ PI 2.0)))
           2.0)
          t_0)
         (- b_m a))
        t_1)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (a + b_m) * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556));
	double t_1 = ((sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0))) * 2.0) * t_0) * (b_m - a);
	double tmp;
	if (b_m <= 1.35e-214) {
		tmp = t_1;
	} else if (b_m <= 5.5e+196) {
		tmp = ((sin((-((0.005555555555555556 * angle_m) * ((double) M_PI)) + (((double) M_PI) / 2.0))) * 2.0) * t_0) * (b_m - a);
	} else {
		tmp = t_1;
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(a + b_m) * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)))
	t_1 = Float64(Float64(Float64(sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0))) * 2.0) * t_0) * Float64(b_m - a))
	tmp = 0.0
	if (b_m <= 1.35e-214)
		tmp = t_1;
	elseif (b_m <= 5.5e+196)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(-Float64(Float64(0.005555555555555556 * angle_m) * pi)) + Float64(pi / 2.0))) * 2.0) * t_0) * Float64(b_m - a));
	else
		tmp = t_1;
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 1.35e-214], t$95$1, If[LessEqual[b$95$m, 5.5e+196], N[(N[(N[(N[Sin[N[((-N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e-214 or 5.49999999999999973e196 < b

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 67.0

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

   10. Applied rewrites 67.0%

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

   if 1.35e-214 < b < 5.49999999999999973e196

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-PI.f64 66.8

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

   10. Applied rewrites 66.8%

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

Alternative 2: 67.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_1 := \left(\left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right) \cdot 2\right) \cdot \left(\left(a + b\_m\right) \cdot \sin t\_0\right)\right) \cdot \left(b\_m - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 1.35 \cdot 10^{-214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b\_m \leq 5.5 \cdot 10^{+196}:\\ \;\;\;\;\left(2 \cdot \sin \left(\left(-t\_0\right) + \frac{\pi}{2}\right)\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.005555555555555556))
        (t_1
         (*
          (*
           (* (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))) 2.0)
           (* (+ a b_m) (sin t_0)))
          (- b_m a))))
   (*
    angle_s
    (if (<= b_m 1.35e-214)
      t_1
      (if (<= b_m 5.5e+196)
        (*
         (* 2.0 (sin (+ (- t_0) (/ PI 2.0))))
         (*
          (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b_m))
          (- b_m a)))
        t_1)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_1 = ((sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0))) * 2.0) * ((a + b_m) * sin(t_0))) * (b_m - a);
	double tmp;
	if (b_m <= 1.35e-214) {
		tmp = t_1;
	} else if (b_m <= 5.5e+196) {
		tmp = (2.0 * sin((-t_0 + (((double) M_PI) / 2.0)))) * ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b_m)) * (b_m - a));
	} else {
		tmp = t_1;
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_1 = Float64(Float64(Float64(sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0))) * 2.0) * Float64(Float64(a + b_m) * sin(t_0))) * Float64(b_m - a))
	tmp = 0.0
	if (b_m <= 1.35e-214)
		tmp = t_1;
	elseif (b_m <= 5.5e+196)
		tmp = Float64(Float64(2.0 * sin(Float64(Float64(-t_0) + Float64(pi / 2.0)))) * Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = t_1;
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[b$95$m, 1.35e-214], t$95$1, If[LessEqual[b$95$m, 5.5e+196], N[(N[(2.0 * N[Sin[N[((-t$95$0) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e-214 or 5.49999999999999973e196 < b

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 67.0

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

   10. Applied rewrites 67.0%

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

   if 1.35e-214 < b < 5.49999999999999973e196

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-sin.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-PI.f64 66.8

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

   8. Applied rewrites 66.8%

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

Alternative 3: 66.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\\ t_2 := -t\_1\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 2\right) \cdot \left(\left(\sin t\_1 \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, t\_2\right)}{2}\right) \cdot \cos \left(\frac{t\_1 - t\_2}{2}\right)\right)\right) \cdot \left(\left(a + b\_m\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b\_m - a\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (* (* 0.005555555555555556 angle_m) PI))
        (t_2 (- t_1)))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         (- INFINITY))
      (*
       (fma (* -3.08641975308642e-5 (* angle_m angle_m)) (* PI PI) 2.0)
       (* (* (sin t_1) (+ a b_m)) (- b_m a)))
      (*
       (*
        (*
         2.0
         (*
          (cos (/ (fma (* 0.005555555555555556 angle_m) PI t_2) 2.0))
          (cos (/ (- t_1 t_2) 2.0))))
        (* (+ a b_m) (sin (* (* angle_m PI) 0.005555555555555556))))
       (- b_m a))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = (0.005555555555555556 * angle_m) * ((double) M_PI);
	double t_2 = -t_1;
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -((double) INFINITY)) {
		tmp = fma((-3.08641975308642e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 2.0) * ((sin(t_1) * (a + b_m)) * (b_m - a));
	} else {
		tmp = ((2.0 * (cos((fma((0.005555555555555556 * angle_m), ((double) M_PI), t_2) / 2.0)) * cos(((t_1 - t_2) / 2.0)))) * ((a + b_m) * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)))) * (b_m - a);
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(0.005555555555555556 * angle_m) * pi)
	t_2 = Float64(-t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= Float64(-Inf))
		tmp = Float64(fma(Float64(-3.08641975308642e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 2.0) * Float64(Float64(sin(t_1) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle_m), pi, t_2) / 2.0)) * cos(Float64(Float64(t_1 - t_2) / 2.0)))) * Float64(Float64(a + b_m) * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)))) * Float64(b_m - a));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 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], (-Infinity)], N[(N[(N[(-3.08641975308642e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$1 - t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -inf.0

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \left(2 + \frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{32400} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      10. lift-PI.f64 62.3

          \[\leadsto \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 2\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 62.3%

      \[\leadsto \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 2\right) \cdot \left(\color{blue}{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-*.f64 N/A

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

   10. Applied rewrites 66.8%

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

Alternative 4: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\\ t_1 := -t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.4 \cdot 10^{+249}:\\ \;\;\;\;\left(\left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right) \cdot 2\right) \cdot \left(\left(a + b\_m\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b\_m - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, t\_1\right)}{2}\right) \cdot \cos \left(\frac{t\_0 - t\_1}{2}\right)\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle_m) PI)) (t_1 (- t_0)))
   (*
    angle_s
    (if (<= angle_m 1.4e+249)
      (*
       (*
        (* (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))) 2.0)
        (* (+ a b_m) (sin (* (* angle_m PI) 0.005555555555555556))))
       (- b_m a))
      (*
       (*
        2.0
        (*
         (cos (/ (fma (* 0.005555555555555556 angle_m) PI t_1) 2.0))
         (cos (/ (- t_0 t_1) 2.0))))
       (*
        (sin (* (* PI angle_m) 0.005555555555555556))
        (* (+ b_m a) (- b_m a))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (0.005555555555555556 * angle_m) * ((double) M_PI);
	double t_1 = -t_0;
	double tmp;
	if (angle_m <= 1.4e+249) {
		tmp = ((sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0))) * 2.0) * ((a + b_m) * sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)))) * (b_m - a);
	} else {
		tmp = (2.0 * (cos((fma((0.005555555555555556 * angle_m), ((double) M_PI), t_1) / 2.0)) * cos(((t_0 - t_1) / 2.0)))) * (sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * ((b_m + a) * (b_m - a)));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(0.005555555555555556 * angle_m) * pi)
	t_1 = Float64(-t_0)
	tmp = 0.0
	if (angle_m <= 1.4e+249)
		tmp = Float64(Float64(Float64(sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0))) * 2.0) * Float64(Float64(a + b_m) * sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)))) * Float64(b_m - a));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(Float64(fma(Float64(0.005555555555555556 * angle_m), pi, t_1) / 2.0)) * cos(Float64(Float64(t_0 - t_1) / 2.0)))) * Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * Float64(Float64(b_m + a) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, N[(angle$95$s * If[LessEqual[angle$95$m, 1.4e+249], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[N[(N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$0 - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.40000000000000009e249

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 67.0

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

   10. Applied rewrites 67.0%

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

   if 1.40000000000000009e249 < angle

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. sum-cos N/A

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

   6. Applied rewrites 57.0%

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

Alternative 5: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 10^{+61}:\\ \;\;\;\;\left(2 \cdot \cos t\_0\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle\_m, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \left(\sin t\_0 \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle_m) 0.005555555555555556)))
   (*
    angle_s
    (if (<= angle_m 1e+61)
      (*
       (* 2.0 (cos t_0))
       (*
        (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b_m))
        (- b_m a)))
      (*
       (* 2.0 (sin (fma (* 0.005555555555555556 angle_m) PI (/ PI 2.0))))
       (* (sin t_0) (* (+ b_m a) (- b_m a))))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (((double) M_PI) * angle_m) * 0.005555555555555556;
	double tmp;
	if (angle_m <= 1e+61) {
		tmp = (2.0 * cos(t_0)) * ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b_m)) * (b_m - a));
	} else {
		tmp = (2.0 * sin(fma((0.005555555555555556 * angle_m), ((double) M_PI), (((double) M_PI) / 2.0)))) * (sin(t_0) * ((b_m + a) * (b_m - a)));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(pi * angle_m) * 0.005555555555555556)
	tmp = 0.0
	if (angle_m <= 1e+61)
		tmp = Float64(Float64(2.0 * cos(t_0)) * Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b_m)) * Float64(b_m - a)));
	else
		tmp = Float64(Float64(2.0 * sin(fma(Float64(0.005555555555555556 * angle_m), pi, Float64(pi / 2.0)))) * Float64(sin(t_0) * Float64(Float64(b_m + a) * Float64(b_m - a))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 1e+61], N[(N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.99999999999999949e60

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   if 9.99999999999999949e60 < angle

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 57.2

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

   6. Applied rewrites 57.2%

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

Alternative 6: 66.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ t_1 := \left(a + b\_m\right) \cdot \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 8.5 \cdot 10^{+213}:\\ \;\;\;\;\left(\left(\cos t\_0 \cdot 2\right) \cdot t\_1\right) \cdot \left(b\_m - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 2\right) \cdot t\_1\right) \cdot \left(b\_m - a\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.005555555555555556))
        (t_1 (* (+ a b_m) (sin t_0))))
   (*
    angle_s
    (if (<= a 8.5e+213)
      (* (* (* (cos t_0) 2.0) t_1) (- b_m a))
      (*
       (* (fma (* -3.08641975308642e-5 (* angle_m angle_m)) (* PI PI) 2.0) t_1)
       (- b_m a))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double t_1 = (a + b_m) * sin(t_0);
	double tmp;
	if (a <= 8.5e+213) {
		tmp = ((cos(t_0) * 2.0) * t_1) * (b_m - a);
	} else {
		tmp = (fma((-3.08641975308642e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 2.0) * t_1) * (b_m - a);
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	t_1 = Float64(Float64(a + b_m) * sin(t_0))
	tmp = 0.0
	if (a <= 8.5e+213)
		tmp = Float64(Float64(Float64(cos(t_0) * 2.0) * t_1) * Float64(b_m - a));
	else
		tmp = Float64(Float64(fma(Float64(-3.08641975308642e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 2.0) * t_1) * Float64(b_m - a));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + b$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[a, 8.5e+213], N[(N[(N[(N[Cos[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-3.08641975308642e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 8.4999999999999995e213

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

   if 8.4999999999999995e213 < a

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(\left(2 + \frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(\left(\frac{-1}{32400} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       5. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       9. lift-PI.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{180}\right)\right)\right) \cdot \left(b - a\right) \]

       10. lift-PI.f64 61.7

           \[\leadsto \left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b - a\right) \]

   11. Applied rewrites 61.7%

       \[\leadsto \left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 2\right) \cdot \left(\left(a + b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b - a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \pi, 2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1
         (*
          (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b_m))
          (- b_m a))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -5e+254)
      (* (fma (* -3.08641975308642e-5 (* angle_m angle_m)) (* PI PI) 2.0) t_1)
      (* 2.0 t_1)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = (sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b_m)) * (b_m - a);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -5e+254) {
		tmp = fma((-3.08641975308642e-5 * (angle_m * angle_m)), (((double) M_PI) * ((double) M_PI)), 2.0) * t_1;
	} else {
		tmp = 2.0 * t_1;
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b_m)) * Float64(b_m - a))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -5e+254)
		tmp = Float64(fma(Float64(-3.08641975308642e-5 * Float64(angle_m * angle_m)), Float64(pi * pi), 2.0) * t_1);
	else
		tmp = Float64(2.0 * t_1);
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 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], -5e+254], N[(N[(N[(-3.08641975308642e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 * t$95$1), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -4.99999999999999994e254

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \left(2 + \frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{32400} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\left(a + b\right)}\right) \cdot \left(b - a\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(\color{blue}{a} + b\right)\right) \cdot \left(b - a\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), {\mathsf{PI}\left(\right)}^{2}, 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + \color{blue}{b}\right)\right) \cdot \left(b - a\right)\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400} \cdot \left(angle \cdot angle\right), \pi \cdot \mathsf{PI}\left(\right), 2\right) \cdot \left(\left(\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

      10. lift-PI.f64 62.3

          \[\leadsto \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 2\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \]

   9. Applied rewrites 62.3%

      \[\leadsto \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), \pi \cdot \pi, 2\right) \cdot \left(\color{blue}{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right) \]

   if -4.99999999999999994e254 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 65.7%

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

Alternative 8: 65.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

2. Taylor expanded in angle around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 57.1%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift--.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. +-commutative N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 67.0%

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

8. Applied rewrites 66.9%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 67.0

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

10. Applied rewrites 67.0%

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

Alternative 9: 65.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;\left(\mathsf{fma}\left(2 \cdot \left(angle\_m \cdot angle\_m\right), \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}, \left(\left(a + b\_m\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right) \cdot \left(b\_m - a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -2e+295)
      (*
       (*
        (fma
         (* 2.0 (* angle_m angle_m))
         (* (* (* (* PI PI) PI) (+ a b_m)) -1.1431184270690443e-7)
         (* (* (+ a b_m) PI) 0.011111111111111112))
        angle_m)
       (- b_m a))
      (*
       2.0
       (*
        (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b_m))
        (- b_m a)))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+295) {
		tmp = (fma((2.0 * (angle_m * angle_m)), ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * (a + b_m)) * -1.1431184270690443e-7), (((a + b_m) * ((double) M_PI)) * 0.011111111111111112)) * angle_m) * (b_m - a);
	} else {
		tmp = 2.0 * ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b_m)) * (b_m - a));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -2e+295)
		tmp = Float64(Float64(fma(Float64(2.0 * Float64(angle_m * angle_m)), Float64(Float64(Float64(Float64(pi * pi) * pi) * Float64(a + b_m)) * -1.1431184270690443e-7), Float64(Float64(Float64(a + b_m) * pi) * 0.011111111111111112)) * angle_m) * Float64(b_m - a));
	else
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b_m)) * Float64(b_m - a)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 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], -2e+295], N[(N[(N[(N[(2.0 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision] + N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -2e295

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right) \]

   11. Applied rewrites 61.0%

       \[\leadsto \left(\mathsf{fma}\left(2 \cdot \left(angle \cdot angle\right), \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}, \left(\left(a + b\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(\color{blue}{b} - a\right) \]

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 65.7%

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

Alternative 10: 65.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := \left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;\left(\mathsf{fma}\left(2 \cdot \left(angle\_m \cdot angle\_m\right), \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}, \left(\left(a + b\_m\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot angle\_m\right) \cdot \left(b\_m - a\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+293}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot \left(a + b\_m\right)\right) \cdot \left(b\_m - a\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))
   (*
    angle_s
    (if (<= t_1 -2e+295)
      (*
       (*
        (fma
         (* 2.0 (* angle_m angle_m))
         (* (* (* (* PI PI) PI) (+ a b_m)) -1.1431184270690443e-7)
         (* (* (+ a b_m) PI) 0.011111111111111112))
        angle_m)
       (- b_m a))
      (if (<= t_1 1e+293)
        (*
         2.0
         (*
          (sin (* (* PI angle_m) 0.005555555555555556))
          (* (+ b_m a) (- b_m a))))
        (*
         (* (* (* angle_m PI) (+ a b_m)) (- b_m a))
         0.011111111111111112))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
	double tmp;
	if (t_1 <= -2e+295) {
		tmp = (fma((2.0 * (angle_m * angle_m)), ((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * (a + b_m)) * -1.1431184270690443e-7), (((a + b_m) * ((double) M_PI)) * 0.011111111111111112)) * angle_m) * (b_m - a);
	} else if (t_1 <= 1e+293) {
		tmp = 2.0 * (sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * ((b_m + a) * (b_m - a)));
	} else {
		tmp = (((angle_m * ((double) M_PI)) * (a + b_m)) * (b_m - a)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
	tmp = 0.0
	if (t_1 <= -2e+295)
		tmp = Float64(Float64(fma(Float64(2.0 * Float64(angle_m * angle_m)), Float64(Float64(Float64(Float64(pi * pi) * pi) * Float64(a + b_m)) * -1.1431184270690443e-7), Float64(Float64(Float64(a + b_m) * pi) * 0.011111111111111112)) * angle_m) * Float64(b_m - a));
	elseif (t_1 <= 1e+293)
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * Float64(Float64(b_m + a) * Float64(b_m - a))));
	else
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * Float64(a + b_m)) * Float64(b_m - a)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(2.0 * N[(N[Power[b$95$m, 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]}, N[(angle$95$s * If[LessEqual[t$95$1, -2e+295], N[(N[(N[(N[(2.0 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision] + N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+293], N[(2.0 * N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -2e295

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)\right)\right) \cdot \left(\color{blue}{b} - a\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(a + b\right)\right)\right)\right)\right) \cdot angle\right) \cdot \left(b - a\right) \]

   11. Applied rewrites 61.0%

       \[\leadsto \left(\mathsf{fma}\left(2 \cdot \left(angle \cdot angle\right), \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}, \left(\left(a + b\right) \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot angle\right) \cdot \left(\color{blue}{b} - a\right) \]

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 9.9999999999999992e292

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 55.6%

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

   if 9.9999999999999992e292 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 62.1

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

   6. Applied rewrites 62.1%

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

Alternative 11: 62.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{-111}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(a + b\_m\right) \cdot \pi\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 5.6e-111)
    (* (* b_m b_m) (sin (* 2.0 (* (* 0.005555555555555556 angle_m) PI))))
    (* (* (* (* (+ a b_m) PI) angle_m) 0.011111111111111112) (- b_m a)))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 5.6e-111) {
		tmp = (b_m * b_m) * sin((2.0 * ((0.005555555555555556 * angle_m) * ((double) M_PI))));
	} else {
		tmp = ((((a + b_m) * ((double) M_PI)) * angle_m) * 0.011111111111111112) * (b_m - a);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (a <= 5.6e-111) {
		tmp = (b_m * b_m) * Math.sin((2.0 * ((0.005555555555555556 * angle_m) * Math.PI)));
	} else {
		tmp = ((((a + b_m) * Math.PI) * angle_m) * 0.011111111111111112) * (b_m - a);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if a <= 5.6e-111:
		tmp = (b_m * b_m) * math.sin((2.0 * ((0.005555555555555556 * angle_m) * math.pi)))
	else:
		tmp = ((((a + b_m) * math.pi) * angle_m) * 0.011111111111111112) * (b_m - a)
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (a <= 5.6e-111)
		tmp = Float64(Float64(b_m * b_m) * sin(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle_m) * pi))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a + b_m) * pi) * angle_m) * 0.011111111111111112) * Float64(b_m - a));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (a <= 5.6e-111)
		tmp = (b_m * b_m) * sin((2.0 * ((0.005555555555555556 * angle_m) * pi)));
	else
		tmp = ((((a + b_m) * pi) * angle_m) * 0.011111111111111112) * (b_m - a);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 5.6e-111], N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 5.5999999999999999e-111

   1. Initial program 53.3%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-cos.f64 N/A

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

   4. Applied rewrites 35.8%

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

   6. Applied rewrites 35.6%

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

   if 5.5999999999999999e-111 < a

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-PI.f64 62.2

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

   11. Applied rewrites 62.2%

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

Alternative 12: 57.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.9 \cdot 10^{+269}:\\ \;\;\;\;\left(\left(\left(\left(a + b\_m\right) \cdot \pi\right) \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot \left(b\_m - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\right) \cdot \left(-a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.9e+269)
    (* (* (* (* (+ a b_m) PI) angle_m) 0.011111111111111112) (- b_m a))
    (* (* (* PI angle_m) (* (+ b_m a) (- a))) 0.011111111111111112))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 1.9e+269) {
		tmp = ((((a + b_m) * ((double) M_PI)) * angle_m) * 0.011111111111111112) * (b_m - a);
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a) * -a)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 1.9e+269) {
		tmp = ((((a + b_m) * Math.PI) * angle_m) * 0.011111111111111112) * (b_m - a);
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a) * -a)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if angle_m <= 1.9e+269:
		tmp = ((((a + b_m) * math.pi) * angle_m) * 0.011111111111111112) * (b_m - a)
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a) * -a)) * 0.011111111111111112
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 1.9e+269)
		tmp = Float64(Float64(Float64(Float64(Float64(a + b_m) * pi) * angle_m) * 0.011111111111111112) * Float64(b_m - a));
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a) * Float64(-a))) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.9e+269)
		tmp = ((((a + b_m) * pi) * angle_m) * 0.011111111111111112) * (b_m - a);
	else
		tmp = ((pi * angle_m) * ((b_m + a) * -a)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.9e+269], N[(N[(N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.89999999999999991e269

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.0%

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-PI.f64 62.2

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

   11. Applied rewrites 62.2%

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

   if 1.89999999999999991e269 < angle

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 54.0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.9

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

   7. Applied rewrites 37.9%

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

Alternative 13: 57.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(b\_m \cdot \left(b\_m - a\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -2e-205)
    (* (* (* (* angle_m PI) -0.011111111111111112) a) a)
    (* (* (* PI angle_m) (* b_m (- b_m a))) 0.011111111111111112))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -2e-205) {
		tmp = (((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((((double) M_PI) * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -2e-205) {
		tmp = (((angle_m * Math.PI) * -0.011111111111111112) * a) * a;
	} else {
		tmp = ((Math.PI * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -2e-205:
		tmp = (((angle_m * math.pi) * -0.011111111111111112) * a) * a
	else:
		tmp = ((math.pi * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-205)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(b_m * Float64(b_m - a))) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-205)
		tmp = (((angle_m * pi) * -0.011111111111111112) * a) * a;
	else
		tmp = ((pi * angle_m) * (b_m * (b_m - a))) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-205], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b$95$m * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -2e-205

   1. Initial program 53.3%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 54.0

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

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   5. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

   7. Applied rewrites 34.3%

      \[\leadsto \mathsf{fma}\left(0.011111111111111112, \mathsf{fma}\left(angle, \frac{\left(0 \cdot b\right) \cdot \pi}{a}, \frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a \cdot a}\right), -0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      4. lift-PI.f64 35.3

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

   10. Applied rewrites 35.3%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot a\right) \cdot a \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot a\right) \cdot a \]

       5. lower-*.f64 38.8

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a \]

   12. Applied rewrites 38.8%

       \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a \]

   if -2e-205 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.0

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.1%

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -2e-205)
    (* (* (* (* angle_m PI) -0.011111111111111112) a) a)
    (* (* 0.011111111111111112 angle_m) (* PI (* b_m b_m))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -2e-205) {
		tmp = (((angle_m * ((double) M_PI)) * -0.011111111111111112) * a) * a;
	} else {
		tmp = (0.011111111111111112 * angle_m) * (((double) M_PI) * (b_m * b_m));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -2e-205) {
		tmp = (((angle_m * Math.PI) * -0.011111111111111112) * a) * a;
	} else {
		tmp = (0.011111111111111112 * angle_m) * (Math.PI * (b_m * b_m));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -2e-205:
		tmp = (((angle_m * math.pi) * -0.011111111111111112) * a) * a
	else:
		tmp = (0.011111111111111112 * angle_m) * (math.pi * (b_m * b_m))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-205)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * -0.011111111111111112) * a) * a);
	else
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(pi * Float64(b_m * b_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-205)
		tmp = (((angle_m * pi) * -0.011111111111111112) * a) * a;
	else
		tmp = (0.011111111111111112 * angle_m) * (pi * (b_m * b_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-205], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$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)))) < -2e-205

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.0

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   5. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

   7. Applied rewrites 34.3%

      \[\leadsto \mathsf{fma}\left(0.011111111111111112, \mathsf{fma}\left(angle, \frac{\left(0 \cdot b\right) \cdot \pi}{a}, \frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a \cdot a}\right), -0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      4. lift-PI.f64 35.3

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

   10. Applied rewrites 35.3%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot a\right) \cdot a \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot a\right) \cdot a \]

       5. lower-*.f64 38.8

          \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a \]

   12. Applied rewrites 38.8%

       \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot a\right) \cdot a \]

   if -2e-205 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.0

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   5. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

   7. Applied rewrites 34.3%

      \[\leadsto \mathsf{fma}\left(0.011111111111111112, \mathsf{fma}\left(angle, \frac{\left(0 \cdot b\right) \cdot \pi}{a}, \frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a \cdot a}\right), -0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      4. lift-PI.f64 35.3

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

   10. Applied rewrites 35.3%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left({b}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left({b}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {b}^{\color{blue}{2}}\right) \]

       5. pow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot \color{blue}{b}\right)\right) \]

       8. lift-PI.f64 34.8

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

   13. Applied rewrites 34.8%

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 49.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -2e-205)
    (* (* (* -0.011111111111111112 angle_m) PI) (* a a))
    (* (* 0.011111111111111112 angle_m) (* PI (* b_m b_m))))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -2e-205) {
		tmp = ((-0.011111111111111112 * angle_m) * ((double) M_PI)) * (a * a);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (((double) M_PI) * (b_m * b_m));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= -2e-205) {
		tmp = ((-0.011111111111111112 * angle_m) * Math.PI) * (a * a);
	} else {
		tmp = (0.011111111111111112 * angle_m) * (Math.PI * (b_m * b_m));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= -2e-205:
		tmp = ((-0.011111111111111112 * angle_m) * math.pi) * (a * a)
	else:
		tmp = (0.011111111111111112 * angle_m) * (math.pi * (b_m * b_m))
	return angle_s * tmp

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-205)
		tmp = Float64(Float64(Float64(-0.011111111111111112 * angle_m) * pi) * Float64(a * a));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(pi * Float64(b_m * b_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= -2e-205)
		tmp = ((-0.011111111111111112 * angle_m) * pi) * (a * a);
	else
		tmp = (0.011111111111111112 * angle_m) * (pi * (b_m * b_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-205], N[(N[(N[(-0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$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)))) < -2e-205

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.0

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   5. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

   7. Applied rewrites 34.3%

      \[\leadsto \mathsf{fma}\left(0.011111111111111112, \mathsf{fma}\left(angle, \frac{\left(0 \cdot b\right) \cdot \pi}{a}, \frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a \cdot a}\right), -0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      4. lift-PI.f64 35.3

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

   10. Applied rewrites 35.3%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

       8. lift-PI.f64 35.3

          \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]

   12. Applied rewrites 35.3%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]

   if -2e-205 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 53.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

      13. lower--.f64 54.0

          \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

   4. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

   5. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

   7. Applied rewrites 34.3%

      \[\leadsto \mathsf{fma}\left(0.011111111111111112, \mathsf{fma}\left(angle, \frac{\left(0 \cdot b\right) \cdot \pi}{a}, \frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a \cdot a}\right), -0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   8. Taylor expanded in a around inf

      \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

      4. lift-PI.f64 35.3

         \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

   10. Applied rewrites 35.3%

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left({b}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left({b}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {b}^{\color{blue}{2}}\right) \]

       5. pow2 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot \color{blue}{b}\right)\right) \]

       8. lift-PI.f64 34.8

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

   13. Applied rewrites 34.8%

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 35.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(-0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a \cdot a\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b_m angle_m)
 :precision binary64
 (* angle_s (* (* (* -0.011111111111111112 angle_m) PI) (* a a))))

C

b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (((-0.011111111111111112 * angle_m) * ((double) M_PI)) * (a * a));
}

Java

b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b_m, double angle_m) {
	return angle_s * (((-0.011111111111111112 * angle_m) * Math.PI) * (a * a));
}

Python

b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b_m, angle_m):
	return angle_s * (((-0.011111111111111112 * angle_m) * math.pi) * (a * a))

Julia

b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b_m, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(-0.011111111111111112 * angle_m) * pi) * Float64(a * a)))
end

MATLAB

b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b_m, angle_m)
	tmp = angle_s * (((-0.011111111111111112 * angle_m) * pi) * (a * a));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(-0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \color{blue}{\frac{1}{90}} \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \frac{1}{90} \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90} \]

   10. difference-of-squares N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90} \]

   13. lower--.f64 54.0

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112 \]

4. Applied rewrites 54.0%

   \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]

5. Taylor expanded in a around inf

   \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{1}{90} \cdot \frac{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)}{a} + \frac{1}{90} \cdot \frac{angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}{{a}^{2}}\right)\right) \cdot {a}^{\color{blue}{2}} \]

7. Applied rewrites 34.3%

   \[\leadsto \mathsf{fma}\left(0.011111111111111112, \mathsf{fma}\left(angle, \frac{\left(0 \cdot b\right) \cdot \pi}{a}, \frac{\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle}{a \cdot a}\right), -0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

8. Taylor expanded in a around inf

   \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

   4. lift-PI.f64 35.3

      \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]

10. Applied rewrites 35.3%

    \[\leadsto \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

    2. lift-PI.f64 N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{90}\right) \cdot \left(a \cdot a\right) \]

    4. *-commutative N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right) \]

    5. associate-*r* N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(a \cdot a\right) \]

    8. lift-PI.f64 35.3

       \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]

12. Applied rewrites 35.3%

    \[\leadsto \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot a\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))))