ab-angle->ABCF B

Percentage Accurate: 54.4% → 68.0%

Time: 8.9s

Alternatives: 16

Speedup: 6.6×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 68.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -5 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\sin \left(\frac{angle\_m}{180} \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{\frac{\mathsf{fma}\left(\pi, -angle\_m, \pi \cdot angle\_m\right)}{-180}}{2}\right) \cdot \cos \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi}, 0.5 \cdot \pi\right)\right) \cdot 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         -5e-65)
      (*
       (* (* (sin (* (/ angle_m 180.0) PI)) (+ a b)) (- b a))
       (*
        2.0
        (*
         (cos (/ (/ (fma PI (- angle_m) (* PI angle_m)) -180.0) 2.0))
         (cos (* (* angle_m 0.005555555555555556) PI)))))
      (*
       (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b)) (- b a))
       (*
        (sin
         (fma
          (* (* angle_m 0.005555555555555556) (cbrt (* PI PI)))
          (cbrt PI)
          (* 0.5 PI)))
        2.0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= -5e-65) {
		tmp = ((sin(((angle_m / 180.0) * ((double) M_PI))) * (a + b)) * (b - a)) * (2.0 * (cos(((fma(((double) M_PI), -angle_m, (((double) M_PI) * angle_m)) / -180.0) / 2.0)) * cos(((angle_m * 0.005555555555555556) * ((double) M_PI)))));
	} else {
		tmp = ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b)) * (b - a)) * (sin(fma(((angle_m * 0.005555555555555556) * cbrt((((double) M_PI) * ((double) M_PI)))), cbrt(((double) M_PI)), (0.5 * ((double) M_PI)))) * 2.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -5e-65)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(angle_m / 180.0) * pi)) * Float64(a + b)) * Float64(b - a)) * Float64(2.0 * Float64(cos(Float64(Float64(fma(pi, Float64(-angle_m), Float64(pi * angle_m)) / -180.0) / 2.0)) * cos(Float64(Float64(angle_m * 0.005555555555555556) * pi)))));
	else
		tmp = Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b)) * Float64(b - a)) * Float64(sin(fma(Float64(Float64(angle_m * 0.005555555555555556) * cbrt(Float64(pi * pi))), cbrt(pi), Float64(0.5 * pi))) * 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], -5e-65], N[(N[(N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[N[(N[(N[(Pi * (-angle$95$m) + N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision] / -180.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $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))))) < -4.99999999999999983e-65

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. div-add-rev N/A

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

   7. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. lower-/.f64 68.0

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

   9. Applied rewrites 68.0%

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

   if -4.99999999999999983e-65 < (*.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 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 67.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

      \[\leadsto \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 2\right) \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. mult-flip N/A

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

      20. lift-/.f64 N/A

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

      21. lift-PI.f64 N/A

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

      22. add-cube-cbrt N/A

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

      23. associate-*r* N/A

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

      24. lower-fma.f64 N/A

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

   7. Applied rewrites 67.8%

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

Alternative 2: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{\frac{\mathsf{fma}\left(\pi, -angle\_m, \pi \cdot angle\_m\right)}{-180}}{2}\right) \cdot \cos \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi}, 0.5 \cdot \pi\right)\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -4e+93)
    (*
     (* (* a (sin (* 0.005555555555555556 (* angle_m PI)))) (- b a))
     (*
      2.0
      (*
       (cos (/ (/ (fma PI (- angle_m) (* PI angle_m)) -180.0) 2.0))
       (cos (* (* angle_m 0.005555555555555556) PI)))))
    (*
     (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b)) (- b a))
     (*
      (sin
       (fma
        (* (* angle_m 0.005555555555555556) (cbrt (* PI PI)))
        (cbrt PI)
        (* 0.5 PI)))
      2.0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -4e+93) {
		tmp = ((a * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))) * (b - a)) * (2.0 * (cos(((fma(((double) M_PI), -angle_m, (((double) M_PI) * angle_m)) / -180.0) / 2.0)) * cos(((angle_m * 0.005555555555555556) * ((double) M_PI)))));
	} else {
		tmp = ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b)) * (b - a)) * (sin(fma(((angle_m * 0.005555555555555556) * cbrt((((double) M_PI) * ((double) M_PI)))), cbrt(((double) M_PI)), (0.5 * ((double) M_PI)))) * 2.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -4e+93)
		tmp = Float64(Float64(Float64(a * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))) * Float64(b - a)) * Float64(2.0 * Float64(cos(Float64(Float64(fma(pi, Float64(-angle_m), Float64(pi * angle_m)) / -180.0) / 2.0)) * cos(Float64(Float64(angle_m * 0.005555555555555556) * pi)))));
	else
		tmp = Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b)) * Float64(b - a)) * Float64(sin(fma(Float64(Float64(angle_m * 0.005555555555555556) * cbrt(Float64(pi * pi))), cbrt(pi), Float64(0.5 * pi))) * 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+93], N[(N[(N[(a * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[N[(N[(N[(Pi * (-angle$95$m) + N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision] / -180.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $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)))) < -4.00000000000000017e93

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. div-add-rev N/A

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 42.2

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

   10. Applied rewrites 42.2%

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

   if -4.00000000000000017e93 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 67.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

      \[\leadsto \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 2\right) \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. mult-flip N/A

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

      20. lift-/.f64 N/A

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

      21. lift-PI.f64 N/A

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

      22. add-cube-cbrt N/A

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

      23. associate-*r* N/A

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

      24. lower-fma.f64 N/A

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

   7. Applied rewrites 67.8%

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

Alternative 3: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sqrt[3]{\pi \cdot \pi}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 8.5 \cdot 10^{+118}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(t\_0 \cdot \left(\sqrt[3]{\pi} \cdot angle\_m\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot t\_0, \sqrt[3]{\pi}, 0.5 \cdot \pi\right)\right) \cdot 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (cbrt (* PI PI))))
   (*
    angle_s
    (if (<= angle_m 8.5e+118)
      (*
       (* (sin (* 0.011111111111111112 (* t_0 (* (cbrt PI) angle_m)))) (- b a))
       (+ b a))
      (*
       (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b)) (- b a))
       (*
        (sin
         (fma (* (* angle_m 0.005555555555555556) t_0) (cbrt PI) (* 0.5 PI)))
        2.0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = cbrt((((double) M_PI) * ((double) M_PI)));
	double tmp;
	if (angle_m <= 8.5e+118) {
		tmp = (sin((0.011111111111111112 * (t_0 * (cbrt(((double) M_PI)) * angle_m)))) * (b - a)) * (b + a);
	} else {
		tmp = ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b)) * (b - a)) * (sin(fma(((angle_m * 0.005555555555555556) * t_0), cbrt(((double) M_PI)), (0.5 * ((double) M_PI)))) * 2.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = cbrt(Float64(pi * pi))
	tmp = 0.0
	if (angle_m <= 8.5e+118)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(t_0 * Float64(cbrt(pi) * angle_m)))) * Float64(b - a)) * Float64(b + a));
	else
		tmp = Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b)) * Float64(b - a)) * Float64(sin(fma(Float64(Float64(angle_m * 0.005555555555555556) * t_0), cbrt(pi), Float64(0.5 * pi))) * 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 8.5e+118], N[(N[(N[Sin[N[(0.011111111111111112 * N[(t$95$0 * N[(N[Power[Pi, 1/3], $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 8.50000000000000033e118

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      8. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      11. lower-cbrt.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \color{blue}{\pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)}\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      17. lower-cbrt.f64 67.7

          \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   9. Applied rewrites 67.7%

      \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   if 8.50000000000000033e118 < angle

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 67.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

      \[\leadsto \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 2\right) \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. mult-flip N/A

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

      20. lift-/.f64 N/A

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

      21. lift-PI.f64 N/A

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

      22. add-cube-cbrt N/A

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

      23. associate-*r* N/A

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

      24. lower-fma.f64 N/A

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

   7. Applied rewrites 67.8%

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

Alternative 4: 67.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 6.4 \cdot 10^{+140}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot angle\_m\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \left|\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 6.4e+140)
    (*
     (*
      (sin (* 0.011111111111111112 (* (cbrt (* PI PI)) (* (cbrt PI) angle_m))))
      (- b a))
     (+ b a))
    (*
     (* (* 2.0 (* (- b a) (+ b a))) (sin (* PI (/ angle_m 180.0))))
     (sin (fma 0.5 PI (fabs (* (* 0.005555555555555556 angle_m) PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 6.4e+140) {
		tmp = (sin((0.011111111111111112 * (cbrt((((double) M_PI) * ((double) M_PI))) * (cbrt(((double) M_PI)) * angle_m)))) * (b - a)) * (b + a);
	} else {
		tmp = ((2.0 * ((b - a) * (b + a))) * sin((((double) M_PI) * (angle_m / 180.0)))) * sin(fma(0.5, ((double) M_PI), fabs(((0.005555555555555556 * angle_m) * ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 6.4e+140)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(cbrt(Float64(pi * pi)) * Float64(cbrt(pi) * angle_m)))) * Float64(b - a)) * Float64(b + a));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(b - a) * Float64(b + a))) * sin(Float64(pi * Float64(angle_m / 180.0)))) * sin(fma(0.5, pi, abs(Float64(Float64(0.005555555555555556 * angle_m) * pi)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 6.4e+140], N[(N[(N[Sin[N[(0.011111111111111112 * N[(N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * Pi + N[Abs[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 6.40000000000000021e140

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      8. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      11. lower-cbrt.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \color{blue}{\pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)}\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      17. lower-cbrt.f64 67.7

          \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   9. Applied rewrites 67.7%

      \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   if 6.40000000000000021e140 < angle

   1. Initial program 54.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fabs.f64 54.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 54.5

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

      15. lift-/.f64 N/A

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

      16. mult-flip N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 54.5

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

   3. Applied rewrites 54.5%

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, \left|\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right|\right)\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.4

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 58.4

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

   5. Applied rewrites 58.4%

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

Alternative 5: 67.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5.5 \cdot 10^{+140}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot angle\_m\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\sin \left(\pi \cdot \mathsf{fma}\left(angle\_m, 0.005555555555555556, 0.5\right)\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 5.5e+140)
    (*
     (*
      (sin (* 0.011111111111111112 (* (cbrt (* PI PI)) (* (cbrt PI) angle_m))))
      (- b a))
     (+ b a))
    (*
     (* (* (sin (* (* 0.005555555555555556 angle_m) PI)) (+ a b)) (- b a))
     (* (sin (* PI (fma angle_m 0.005555555555555556 0.5))) 2.0)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 5.5e+140) {
		tmp = (sin((0.011111111111111112 * (cbrt((((double) M_PI) * ((double) M_PI))) * (cbrt(((double) M_PI)) * angle_m)))) * (b - a)) * (b + a);
	} else {
		tmp = ((sin(((0.005555555555555556 * angle_m) * ((double) M_PI))) * (a + b)) * (b - a)) * (sin((((double) M_PI) * fma(angle_m, 0.005555555555555556, 0.5))) * 2.0);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 5.5e+140)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(cbrt(Float64(pi * pi)) * Float64(cbrt(pi) * angle_m)))) * Float64(b - a)) * Float64(b + a));
	else
		tmp = Float64(Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle_m) * pi)) * Float64(a + b)) * Float64(b - a)) * Float64(sin(Float64(pi * fma(angle_m, 0.005555555555555556, 0.5))) * 2.0));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 5.5e+140], N[(N[(N[Sin[N[(0.011111111111111112 * N[(N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 5.5e140

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      8. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      11. lower-cbrt.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \color{blue}{\pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)}\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      17. lower-cbrt.f64 67.7

          \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   9. Applied rewrites 67.7%

      \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   if 5.5e140 < angle

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 67.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

      \[\leadsto \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, \pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 2\right) \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 67.7

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

   7. Applied rewrites 67.7%

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

Alternative 6: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot angle\_m\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 1.5e+154)
    (*
     (*
      (sin (* 0.011111111111111112 (* (cbrt (* PI PI)) (* (cbrt PI) angle_m))))
      (- b a))
     (+ b a))
    (* (* 0.011111111111111112 (- b a)) (* (* (+ b a) PI) angle_m)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.5e+154) {
		tmp = (sin((0.011111111111111112 * (cbrt((((double) M_PI) * ((double) M_PI))) * (cbrt(((double) M_PI)) * angle_m)))) * (b - a)) * (b + a);
	} else {
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.5e+154) {
		tmp = (Math.sin((0.011111111111111112 * (Math.cbrt((Math.PI * Math.PI)) * (Math.cbrt(Math.PI) * angle_m)))) * (b - a)) * (b + a);
	} else {
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 1.5e+154)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(cbrt(Float64(pi * pi)) * Float64(cbrt(pi) * angle_m)))) * Float64(b - a)) * Float64(b + a));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(b - a)) * Float64(Float64(Float64(b + a) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 1.5e+154], N[(N[(N[Sin[N[(0.011111111111111112 * N[(N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 1.50000000000000013e154

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot angle\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      8. cbrt-unprod N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      11. lower-cbrt.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \color{blue}{\pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\color{blue}{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot angle\right)}\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \left(\sin \left(\frac{1}{90} \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

      17. lower-cbrt.f64 67.7

          \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot angle\right)\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   9. Applied rewrites 67.7%

      \[\leadsto \left(\sin \left(0.011111111111111112 \cdot \color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot angle\right)\right)}\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right) \]

   if 1.50000000000000013e154 < a

   1. Initial program 54.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.2

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.2

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.1

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

   6. Applied rewrites 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.7

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

   10. Applied rewrites 62.7%

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

Alternative 7: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \pi\right) \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\left(0.011111111111111112 \cdot angle\_m\right) \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) -2e+301)
    (* (* 0.011111111111111112 (- b a)) (* (* (+ b a) PI) angle_m))
    (* (* (sin (* (* 0.011111111111111112 angle_m) PI)) (- b a)) (+ b a)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= -2e+301) {
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * ((double) M_PI)) * angle_m);
	} else {
		tmp = (sin(((0.011111111111111112 * angle_m) * ((double) M_PI))) * (b - a)) * (b + a);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= -2e+301) {
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * Math.PI) * angle_m);
	} else {
		tmp = (Math.sin(((0.011111111111111112 * angle_m) * Math.PI)) * (b - a)) * (b + a);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= -2e+301:
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * math.pi) * angle_m)
	else:
		tmp = (math.sin(((0.011111111111111112 * angle_m) * math.pi)) * (b - a)) * (b + a)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= -2e+301)
		tmp = Float64(Float64(0.011111111111111112 * Float64(b - a)) * Float64(Float64(Float64(b + a) * pi) * angle_m));
	else
		tmp = Float64(Float64(sin(Float64(Float64(0.011111111111111112 * angle_m) * pi)) * Float64(b - a)) * Float64(b + a));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= -2e+301)
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * pi) * angle_m);
	else
		tmp = (sin(((0.011111111111111112 * angle_m) * pi)) * (b - a)) * (b + a);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+301], N[(N[(0.011111111111111112 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $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)))) < -2.00000000000000011e301

   1. Initial program 54.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.2

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.2

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.1

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

   6. Applied rewrites 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.7

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

   10. Applied rewrites 62.7%

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

   if -2.00000000000000011e301 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.6

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

   9. Applied rewrites 67.6%

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

Alternative 8: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot angle\_m\right)\right) \cdot \left(b + a\right)\right) \cdot \pi\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
         2e+214)
      (* (* (sin (* 0.011111111111111112 (* PI angle_m))) (- b a)) (+ b a))
      (* (* (* 0.011111111111111112 (* (- b a) angle_m)) (+ b a)) PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 2e+214) {
		tmp = (sin((0.011111111111111112 * (((double) M_PI) * angle_m))) * (b - a)) * (b + a);
	} else {
		tmp = ((0.011111111111111112 * ((b - a) * angle_m)) * (b + a)) * ((double) M_PI);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 2e+214) {
		tmp = (Math.sin((0.011111111111111112 * (Math.PI * angle_m))) * (b - a)) * (b + a);
	} else {
		tmp = ((0.011111111111111112 * ((b - a) * angle_m)) * (b + a)) * Math.PI;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (angle_m / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= 2e+214:
		tmp = (math.sin((0.011111111111111112 * (math.pi * angle_m))) * (b - a)) * (b + a)
	else:
		tmp = ((0.011111111111111112 * ((b - a) * angle_m)) * (b + a)) * math.pi
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 2e+214)
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle_m))) * Float64(b - a)) * Float64(b + a));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * Float64(Float64(b - a) * angle_m)) * Float64(b + a)) * pi);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (angle_m / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 2e+214)
		tmp = (sin((0.011111111111111112 * (pi * angle_m))) * (b - a)) * (b + a);
	else
		tmp = ((0.011111111111111112 * ((b - a) * angle_m)) * (b + a)) * pi;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2e+214], N[(N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * N[(N[(b - a), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * Pi), $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))))) < 1.9999999999999999e214

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.6%

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

   if 1.9999999999999999e214 < (*.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 54.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.2

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.2

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.1

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

   6. Applied rewrites 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 62.6

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

   8. Applied rewrites 62.6%

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

Alternative 9: 67.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 67.6%

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

5. Applied rewrites 67.8%

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

7. Applied rewrites 67.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 67.5

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

9. Applied rewrites 67.5%

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

Alternative 10: 65.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-178}:\\ \;\;\;\;\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \pi\right) \cdot angle\_m\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow a 2.0) 2e-178)
    (* (* b (sin (* 0.011111111111111112 (* angle_m PI)))) (+ b a))
    (* (* 0.011111111111111112 (- b a)) (* (* (+ b a) PI) angle_m)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(a, 2.0) <= 2e-178) {
		tmp = (b * sin((0.011111111111111112 * (angle_m * ((double) M_PI))))) * (b + a);
	} else {
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * ((double) M_PI)) * angle_m);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (Math.pow(a, 2.0) <= 2e-178) {
		tmp = (b * Math.sin((0.011111111111111112 * (angle_m * Math.PI)))) * (b + a);
	} else {
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * Math.PI) * angle_m);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if math.pow(a, 2.0) <= 2e-178:
		tmp = (b * math.sin((0.011111111111111112 * (angle_m * math.pi)))) * (b + a)
	else:
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * math.pi) * angle_m)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if ((a ^ 2.0) <= 2e-178)
		tmp = Float64(Float64(b * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))) * Float64(b + a));
	else
		tmp = Float64(Float64(0.011111111111111112 * Float64(b - a)) * Float64(Float64(Float64(b + a) * pi) * angle_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((a ^ 2.0) <= 2e-178)
		tmp = (b * sin((0.011111111111111112 * (angle_m * pi)))) * (b + a);
	else
		tmp = (0.011111111111111112 * (b - a)) * (((b + a) * pi) * angle_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 2e-178], N[(N[(b * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a #s(literal 2 binary64)) < 1.9999999999999999e-178

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 42.7

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

   10. Applied rewrites 42.7%

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

   if 1.9999999999999999e-178 < (pow.f64 a #s(literal 2 binary64))

   1. Initial program 54.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.2

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.2

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 55.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 55.1

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

   6. Applied rewrites 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.7

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

   10. Applied rewrites 62.7%

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

Alternative 11: 62.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 51.2

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

4. Applied rewrites 51.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 51.2

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

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

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 55.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 55.1

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

6. Applied rewrites 55.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 62.6

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

8. Applied rewrites 62.6%

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

Alternative 12: 62.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 51.2

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

4. Applied rewrites 51.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 51.2

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

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

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 55.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 55.1

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

6. Applied rewrites 55.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 62.5

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

8. Applied rewrites 62.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 62.7

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

10. Applied rewrites 62.7%

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

Alternative 13: 62.5% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 51.2

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

4. Applied rewrites 51.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 51.2

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

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

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 55.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 55.1

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

6. Applied rewrites 55.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 62.5

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

8. Applied rewrites 62.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \pi\right) \cdot \color{blue}{angle}\right)\right) \]

   6. lower-*.f64 62.5

      \[\leadsto 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\left(\left(b + a\right) \cdot \pi\right) \cdot \color{blue}{angle}\right)\right) \]

10. Applied rewrites 62.5%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(\left(\left(b + a\right) \cdot \pi\right) \cdot angle\right)}\right) \]
11. Add Preprocessing

Alternative 14: 62.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\_m\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* (- b a) angle_m) (* (+ b a) PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b - a) * angle_m) * ((b + a) * ((double) M_PI))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b - a) * angle_m) * ((b + a) * Math.PI)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (((b - a) * angle_m) * ((b + a) * math.pi)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(Float64(b - a) * angle_m) * Float64(Float64(b + a) * pi))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (((b - a) * angle_m) * ((b + a) * pi)));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(N[(b - a), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 51.2

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 51.2

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. pow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   17. lower-*.f64 55.1

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   20. lower-+.f64 55.1

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 55.1%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

   9. lower-*.f64 62.5

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\pi}\right)\right) \]

8. Applied rewrites 62.5%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]
9. Add Preprocessing

Alternative 15: 61.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b - a\right) \cdot angle\_m\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 2 \cdot 10^{-248}:\\ \;\;\;\;0.011111111111111112 \cdot \left(t\_0 \cdot \left(a \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(t\_0 \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (- b a) angle_m)))
   (*
    angle_s
    (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 2e-248)
      (* 0.011111111111111112 (* t_0 (* a PI)))
      (* 0.011111111111111112 (* t_0 (* b PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b - a) * angle_m;
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 2e-248) {
		tmp = 0.011111111111111112 * (t_0 * (a * ((double) M_PI)));
	} else {
		tmp = 0.011111111111111112 * (t_0 * (b * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b - a) * angle_m;
	double tmp;
	if ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 2e-248) {
		tmp = 0.011111111111111112 * (t_0 * (a * Math.PI));
	} else {
		tmp = 0.011111111111111112 * (t_0 * (b * Math.PI));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b - a) * angle_m
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 2e-248:
		tmp = 0.011111111111111112 * (t_0 * (a * math.pi))
	else:
		tmp = 0.011111111111111112 * (t_0 * (b * math.pi))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b - a) * angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 2e-248)
		tmp = Float64(0.011111111111111112 * Float64(t_0 * Float64(a * pi)));
	else
		tmp = Float64(0.011111111111111112 * Float64(t_0 * Float64(b * pi)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (b - a) * angle_m;
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 2e-248)
		tmp = 0.011111111111111112 * (t_0 * (a * pi));
	else
		tmp = 0.011111111111111112 * (t_0 * (b * pi));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b - a), $MachinePrecision] * angle$95$m), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-248], N[(0.011111111111111112 * N[(t$95$0 * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(t$95$0 * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.99999999999999996e-248

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 51.2

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 51.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. pow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 55.1

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 55.1

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 55.1%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

      9. lower-*.f64 62.5

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\pi}\right)\right) \]

   8. Applied rewrites 62.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. lower-PI.f64 41.4

          \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \pi\right)\right) \]

   11. Applied rewrites 41.4%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \color{blue}{\pi}\right)\right) \]

   if 1.99999999999999996e-248 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

      7. lower-pow.f64 51.2

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

      6. lower-*.f64 51.2

         \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

      9. pow2 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

      12. difference-of-squares-rev N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      17. lower-*.f64 55.1

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

      19. +-commutative N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      20. lower-+.f64 55.1

          \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   6. Applied rewrites 55.1%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

      9. lower-*.f64 62.5

         \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\pi}\right)\right) \]

   8. Applied rewrites 62.5%

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(b \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \]

       2. lower-PI.f64 42.1

          \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(b \cdot \pi\right)\right) \]

   11. Applied rewrites 42.1%

       \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(b \cdot \color{blue}{\pi}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 41.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\_m\right) \cdot \left(a \cdot \pi\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* (- b a) angle_m) (* a PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b - a) * angle_m) * (a * ((double) M_PI))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (((b - a) * angle_m) * (a * Math.PI)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (((b - a) * angle_m) * (a * math.pi)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(Float64(b - a) * angle_m) * Float64(a * pi))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (((b - a) * angle_m) * (a * pi)));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(N[(b - a), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {\color{blue}{a}}^{2}\right)\right)\right) \]

   7. lower-pow.f64 51.2

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{\color{blue}{2}}\right)\right)\right) \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \color{blue}{\pi}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\pi}\right) \]

   6. lower-*.f64 51.2

      \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \pi\right) \]

   9. pow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - {a}^{2}\right)\right) \cdot \pi\right) \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \pi\right) \]

   12. difference-of-squares-rev N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   13. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   15. lift--.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) \cdot \pi\right) \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   17. lower-*.f64 55.1

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot \pi\right) \]

   19. +-commutative N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   20. lower-+.f64 55.1

       \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

6. Applied rewrites 55.1%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \color{blue}{\pi}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right) \cdot \pi\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(b + a\right)\right) \cdot \pi\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \pi\right)\right) \]

   9. lower-*.f64 62.5

      \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\pi}\right)\right) \]

8. Applied rewrites 62.5%

   \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \pi\right)}\right) \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{1}{90} \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \]

    2. lower-PI.f64 41.4

       \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \pi\right)\right) \]

11. Applied rewrites 41.4%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(\left(b - a\right) \cdot angle\right) \cdot \left(a \cdot \color{blue}{\pi}\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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)))))