ab-angle->ABCF B

Percentage Accurate: 54.0% → 67.6%

Time: 15.6s

Alternatives: 16

Speedup: 10.3×

• 33.2% 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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.08 \cdot 10^{+170}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a\_m \leq 3.3 \cdot 10^{+205}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{b\_m}, \sqrt{b\_m}, a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b\_m - a\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b\_m - a\_m\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= a_m 1.08e+170)
   (* (+ b_m a_m) (* (- b_m a_m) (sin (* angle (* 0.011111111111111112 PI)))))
   (if (<= a_m 3.3e+205)
     (* (+ b_m a_m) (* (- b_m a_m) (* 0.011111111111111112 (* angle PI))))
     (*
      (fma (sqrt b_m) (sqrt b_m) a_m)
      (*
       angle
       (fma
        (* -2.2862368541380886e-7 (* angle angle))
        (* (- b_m a_m) (* PI (* PI PI)))
        (* 0.011111111111111112 (* (- b_m a_m) PI))))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (a_m <= 1.08e+170) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((angle * (0.011111111111111112 * ((double) M_PI)))));
	} else if (a_m <= 3.3e+205) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * ((double) M_PI))));
	} else {
		tmp = fma(sqrt(b_m), sqrt(b_m), a_m) * (angle * fma((-2.2862368541380886e-7 * (angle * angle)), ((b_m - a_m) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))), (0.011111111111111112 * ((b_m - a_m) * ((double) M_PI)))));
	}
	return tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (a_m <= 1.08e+170)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(angle * Float64(0.011111111111111112 * pi)))));
	elseif (a_m <= 3.3e+205)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(0.011111111111111112 * Float64(angle * pi))));
	else
		tmp = Float64(fma(sqrt(b_m), sqrt(b_m), a_m) * Float64(angle * fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(Float64(b_m - a_m) * Float64(pi * Float64(pi * pi))), Float64(0.011111111111111112 * Float64(Float64(b_m - a_m) * pi)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[a$95$m, 1.08e+170], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle * N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 3.3e+205], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[b$95$m], $MachinePrecision] * N[Sqrt[b$95$m], $MachinePrecision] + a$95$m), $MachinePrecision] * N[(angle * N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.07999999999999996e170

   1. Initial program 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 61.5

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

   6. Applied rewrites 61.5%

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

   if 1.07999999999999996e170 < a < 3.3000000000000002e205

   1. Initial program 21.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if 3.3000000000000002e205 < a

   1. Initial program 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.6%

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

      1. lift-+.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-fma.f64 27.7

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

   6. Applied rewrites 27.7%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right) + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right) \cdot {\mathsf{PI}\left(\right)}^{3}}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\left(b - a\right)} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      16. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      17. lower-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), \frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   9. Applied rewrites 16.6%

      \[\leadsto \mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+170}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+205}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{b}, \sqrt{b}, a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \left(b - a\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a\_m}^{2} \leq -5 \cdot 10^{-182}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (- (pow b_m 2.0) (pow a_m 2.0)) -5e-182)
   (* a_m (* -0.011111111111111112 (* a_m (* angle PI))))
   (* 0.011111111111111112 (* angle (* PI (* b_m b_m))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((pow(b_m, 2.0) - pow(a_m, 2.0)) <= -5e-182) {
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b_m * b_m)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0)) <= -5e-182) {
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b_m * b_m)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a_m, 2.0)) <= -5e-182:
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b_m * b_m)))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a_m ^ 2.0)) <= -5e-182)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(a_m * Float64(angle * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b_m * b_m))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a_m ^ 2.0)) <= -5e-182)
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * pi)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b_m * b_m)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -5e-182], N[(a$95$m * N[(-0.011111111111111112 * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -5.00000000000000024e-182

   1. Initial program 42.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 42.1

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 40.8%

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

   10. Applied rewrites 55.7%

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

   if -5.00000000000000024e-182 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 55.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 56.8

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

   5. Applied rewrites 56.8%

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

   7. Applied rewrites 15.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 59.1%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-182}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a\_m}^{2} \leq -5 \cdot 10^{-182}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (- (pow b_m 2.0) (pow a_m 2.0)) -5e-182)
   (* a_m (* -0.011111111111111112 (* a_m (* angle PI))))
   (* (* angle 0.011111111111111112) (* PI (* b_m b_m)))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((pow(b_m, 2.0) - pow(a_m, 2.0)) <= -5e-182) {
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * ((double) M_PI))));
	} else {
		tmp = (angle * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0)) <= -5e-182) {
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * Math.PI)));
	} else {
		tmp = (angle * 0.011111111111111112) * (Math.PI * (b_m * b_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a_m, 2.0)) <= -5e-182:
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * math.pi)))
	else:
		tmp = (angle * 0.011111111111111112) * (math.pi * (b_m * b_m))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a_m ^ 2.0)) <= -5e-182)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(a_m * Float64(angle * pi))));
	else
		tmp = Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a_m ^ 2.0)) <= -5e-182)
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * pi)));
	else
		tmp = (angle * 0.011111111111111112) * (pi * (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -5e-182], N[(a$95$m * N[(-0.011111111111111112 * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -5.00000000000000024e-182

   1. Initial program 42.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 42.1

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 40.8%

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

   10. Applied rewrites 55.7%

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

   if -5.00000000000000024e-182 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 55.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 59.0%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-182}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5e+134)
   (* (+ b_m a_m) (* (- b_m a_m) (sin (* PI (* angle 0.011111111111111112)))))
   (*
    (* (* 0.011111111111111112 (* angle PI)) (* (+ b_m a_m) (- b_m a_m)))
    (cos (* (/ angle 180.0) PI)))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e+134) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((((double) M_PI) * (angle * 0.011111111111111112))));
	} else {
		tmp = ((0.011111111111111112 * (angle * ((double) M_PI))) * ((b_m + a_m) * (b_m - a_m))) * cos(((angle / 180.0) * ((double) M_PI)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5e+134) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((Math.PI * (angle * 0.011111111111111112))));
	} else {
		tmp = ((0.011111111111111112 * (angle * Math.PI)) * ((b_m + a_m) * (b_m - a_m))) * Math.cos(((angle / 180.0) * Math.PI));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 5e+134:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin((math.pi * (angle * 0.011111111111111112))))
	else:
		tmp = ((0.011111111111111112 * (angle * math.pi)) * ((b_m + a_m) * (b_m - a_m))) * math.cos(((angle / 180.0) * math.pi))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e+134)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(angle * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(Float64(0.011111111111111112 * Float64(angle * pi)) * Float64(Float64(b_m + a_m) * Float64(b_m - a_m))) * cos(Float64(Float64(angle / 180.0) * pi)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 5e+134)
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((pi * (angle * 0.011111111111111112))));
	else
		tmp = ((0.011111111111111112 * (angle * pi)) * ((b_m + a_m) * (b_m - a_m))) * cos(((angle / 180.0) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+134], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999981e134

   1. Initial program 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.9

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

   6. Applied rewrites 69.9%

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

   if 4.99999999999999981e134 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 23.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      12. lower--.f64 38.8

          \[\leadsto \left(\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 2 \cdot 10^{-183}:\\ \;\;\;\;\left(a\_m \cdot \left(angle \cdot a\_m\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (pow b_m 2.0) 2e-183)
   (* (* a_m (* angle a_m)) (* PI -0.011111111111111112))
   (*
    (+ b_m a_m)
    (*
     (- b_m a_m)
     (*
      angle
      (fma
       (* -2.2862368541380886e-7 (* angle angle))
       (* PI (* PI PI))
       (* 0.011111111111111112 PI)))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (pow(b_m, 2.0) <= 2e-183) {
		tmp = (a_m * (angle * a_m)) * (((double) M_PI) * -0.011111111111111112);
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * (angle * fma((-2.2862368541380886e-7 * (angle * angle)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (0.011111111111111112 * ((double) M_PI)))));
	}
	return tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 2e-183)
		tmp = Float64(Float64(a_m * Float64(angle * a_m)) * Float64(pi * -0.011111111111111112));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle * fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(pi * Float64(pi * pi)), Float64(0.011111111111111112 * pi)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 2e-183], N[(N[(a$95$m * N[(angle * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle * N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2.00000000000000001e-183

   1. Initial program 50.1%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.1%

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

   10. Applied rewrites 58.1%

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

   if 2.00000000000000001e-183 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      16. lower-PI.f64 63.9

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Applied rewrites 63.9%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{-183}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-57}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m + a\_m\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1e-57)
   (*
    (+ b_m a_m)
    (*
     (- b_m a_m)
     (*
      angle
      (fma
       (* -2.2862368541380886e-7 (* angle angle))
       (* PI (* PI PI))
       (* 0.011111111111111112 PI)))))
   (*
    (* (+ b_m a_m) (- b_m a_m))
    (sin (* 0.011111111111111112 (* angle PI))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1e-57) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (angle * fma((-2.2862368541380886e-7 * (angle * angle)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (0.011111111111111112 * ((double) M_PI)))));
	} else {
		tmp = ((b_m + a_m) * (b_m - a_m)) * sin((0.011111111111111112 * (angle * ((double) M_PI))));
	}
	return tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-57)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle * fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(pi * Float64(pi * pi)), Float64(0.011111111111111112 * pi)))));
	else
		tmp = Float64(Float64(Float64(b_m + a_m) * Float64(b_m - a_m)) * sin(Float64(0.011111111111111112 * Float64(angle * pi))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-57], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle * N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999955e-58

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      16. lower-PI.f64 70.7

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Applied rewrites 70.7%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]

   if 9.99999999999999955e-58 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 34.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

   4. Applied rewrites 37.1%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-57}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 2.25 \cdot 10^{-91}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(t\_0 \cdot \left(-a\_m\right)\right)\\ \mathbf{elif}\;b\_m \leq 7 \cdot 10^{+220}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(b\_m \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* angle PI)))))
   (if (<= b_m 2.25e-91)
     (* (+ b_m a_m) (* t_0 (- a_m)))
     (if (<= b_m 7e+220)
       (*
        (+ b_m a_m)
        (*
         (- b_m a_m)
         (*
          angle
          (fma
           (* -2.2862368541380886e-7 (* angle angle))
           (* PI (* PI PI))
           (* 0.011111111111111112 PI)))))
       (* (+ b_m a_m) (* b_m t_0))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double t_0 = sin((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 2.25e-91) {
		tmp = (b_m + a_m) * (t_0 * -a_m);
	} else if (b_m <= 7e+220) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (angle * fma((-2.2862368541380886e-7 * (angle * angle)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (0.011111111111111112 * ((double) M_PI)))));
	} else {
		tmp = (b_m + a_m) * (b_m * t_0);
	}
	return tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	t_0 = sin(Float64(0.011111111111111112 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 2.25e-91)
		tmp = Float64(Float64(b_m + a_m) * Float64(t_0 * Float64(-a_m)));
	elseif (b_m <= 7e+220)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle * fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(pi * Float64(pi * pi)), Float64(0.011111111111111112 * pi)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(b_m * t_0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := Block[{t$95$0 = N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 2.25e-91], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(t$95$0 * (-a$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 7e+220], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle * N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.24999999999999988e-91

   1. Initial program 48.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 61.7%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 44.8

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

   7. Applied rewrites 44.8%

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

   if 2.24999999999999988e-91 < b < 6.99999999999999972e220

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      16. lower-PI.f64 58.4

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Applied rewrites 58.4%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]

   if 6.99999999999999972e220 < b

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 93.2

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

   7. Applied rewrites 93.2%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-91}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-a\right)\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+220}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.4 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, \mathsf{fma}\left(0.011111111111111112, \left(angle \cdot \pi\right) \cdot 0, -0.011111111111111112 \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right), \left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 7 \cdot 10^{+220}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(b\_m \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= b_m 2.4e-87)
   (fma
    a_m
    (fma
     0.011111111111111112
     (* (* angle PI) 0.0)
     (* -0.011111111111111112 (* a_m (* angle PI))))
    (* (* angle 0.011111111111111112) (* PI (* b_m b_m))))
   (if (<= b_m 7e+220)
     (*
      (+ b_m a_m)
      (*
       (- b_m a_m)
       (*
        angle
        (fma
         (* -2.2862368541380886e-7 (* angle angle))
         (* PI (* PI PI))
         (* 0.011111111111111112 PI)))))
     (* (+ b_m a_m) (* b_m (sin (* 0.011111111111111112 (* angle PI))))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (b_m <= 2.4e-87) {
		tmp = fma(a_m, fma(0.011111111111111112, ((angle * ((double) M_PI)) * 0.0), (-0.011111111111111112 * (a_m * (angle * ((double) M_PI))))), ((angle * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m))));
	} else if (b_m <= 7e+220) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (angle * fma((-2.2862368541380886e-7 * (angle * angle)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (0.011111111111111112 * ((double) M_PI)))));
	} else {
		tmp = (b_m + a_m) * (b_m * sin((0.011111111111111112 * (angle * ((double) M_PI)))));
	}
	return tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (b_m <= 2.4e-87)
		tmp = fma(a_m, fma(0.011111111111111112, Float64(Float64(angle * pi) * 0.0), Float64(-0.011111111111111112 * Float64(a_m * Float64(angle * pi)))), Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m))));
	elseif (b_m <= 7e+220)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle * fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(pi * Float64(pi * pi)), Float64(0.011111111111111112 * pi)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(b_m * sin(Float64(0.011111111111111112 * Float64(angle * pi)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[b$95$m, 2.4e-87], N[(a$95$m * N[(0.011111111111111112 * N[(N[(angle * Pi), $MachinePrecision] * 0.0), $MachinePrecision] + N[(-0.011111111111111112 * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 7e+220], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle * N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.4e-87

   1. Initial program 49.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 57.0%

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

   if 2.4e-87 < b < 6.99999999999999972e220

   1. Initial program 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 58.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      16. lower-PI.f64 57.8

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Applied rewrites 57.8%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]

   if 6.99999999999999972e220 < b

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 93.2

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

   7. Applied rewrites 93.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(0.011111111111111112, \left(angle \cdot \pi\right) \cdot 0, -0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right), \left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+220}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.4 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(a\_m, \mathsf{fma}\left(0.011111111111111112, \left(angle \cdot \pi\right) \cdot 0, -0.011111111111111112 \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right), \left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= b_m 2.4e-87)
   (fma
    a_m
    (fma
     0.011111111111111112
     (* (* angle PI) 0.0)
     (* -0.011111111111111112 (* a_m (* angle PI))))
    (* (* angle 0.011111111111111112) (* PI (* b_m b_m))))
   (*
    (+ b_m a_m)
    (*
     (- b_m a_m)
     (*
      angle
      (fma
       (* -2.2862368541380886e-7 (* angle angle))
       (* PI (* PI PI))
       (* 0.011111111111111112 PI)))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if (b_m <= 2.4e-87) {
		tmp = fma(a_m, fma(0.011111111111111112, ((angle * ((double) M_PI)) * 0.0), (-0.011111111111111112 * (a_m * (angle * ((double) M_PI))))), ((angle * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m))));
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * (angle * fma((-2.2862368541380886e-7 * (angle * angle)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (0.011111111111111112 * ((double) M_PI)))));
	}
	return tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (b_m <= 2.4e-87)
		tmp = fma(a_m, fma(0.011111111111111112, Float64(Float64(angle * pi) * 0.0), Float64(-0.011111111111111112 * Float64(a_m * Float64(angle * pi)))), Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle * fma(Float64(-2.2862368541380886e-7 * Float64(angle * angle)), Float64(pi * Float64(pi * pi)), Float64(0.011111111111111112 * pi)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[b$95$m, 2.4e-87], N[(a$95$m * N[(0.011111111111111112 * N[(N[(angle * Pi), $MachinePrecision] * 0.0), $MachinePrecision] + N[(-0.011111111111111112 * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle * N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4e-87

   1. Initial program 49.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 57.0%

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

   if 2.4e-87 < b

   1. Initial program 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{4374000} \cdot {angle}^{2}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{4374000} \cdot {angle}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \color{blue}{\left(angle \cdot angle\right)}, {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-PI.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(\frac{-1}{4374000} \cdot \left(angle \cdot angle\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\frac{1}{90} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      16. lower-PI.f64 63.7

          \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Applied rewrites 63.7%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(0.011111111111111112, \left(angle \cdot \pi\right) \cdot 0, -0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right), \left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi + \pi \cdot 0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+177)
   (* (+ b_m a_m) (* (- b_m a_m) (* 0.011111111111111112 (* angle PI))))
   (* (* b_m b_m) (* 0.011111111111111112 (* angle (+ PI (* PI 0.0)))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * ((double) M_PI))));
	} else {
		tmp = (b_m * b_m) * (0.011111111111111112 * (angle * (((double) M_PI) + (((double) M_PI) * 0.0))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * Math.PI)));
	} else {
		tmp = (b_m * b_m) * (0.011111111111111112 * (angle * (Math.PI + (Math.PI * 0.0))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+177:
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * math.pi)))
	else:
		tmp = (b_m * b_m) * (0.011111111111111112 * (angle * (math.pi + (math.pi * 0.0))))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+177)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(0.011111111111111112 * Float64(angle * pi))));
	else
		tmp = Float64(Float64(b_m * b_m) * Float64(0.011111111111111112 * Float64(angle * Float64(pi + Float64(pi * 0.0)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+177)
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * pi)));
	else
		tmp = (b_m * b_m) * (0.011111111111111112 * (angle * (pi + (pi * 0.0))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+177], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi + N[(Pi * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e177

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 62.8

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

   7. Applied rewrites 62.8%

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

   if 2e177 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 31.7

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

   5. Applied rewrites 31.7%

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 38.9%

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi + \pi \cdot 0\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+177)
   (* (+ b_m a_m) (* (- b_m a_m) (* 0.011111111111111112 (* angle PI))))
   (* 0.011111111111111112 (* angle (* PI (* b_m b_m))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b_m * b_m)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b_m * b_m)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+177:
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b_m * b_m)))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+177)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(0.011111111111111112 * Float64(angle * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b_m * b_m))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+177)
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle * pi)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b_m * b_m)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+177], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e177

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 62.8

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

   7. Applied rewrites 62.8%

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

   if 2e177 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 31.7

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

   5. Applied rewrites 31.7%

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 38.9%

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

Alternative 12: 62.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(\left(b\_m - a\_m\right) \cdot \pi\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+177)
   (* (+ b_m a_m) (* (* (- b_m a_m) PI) (* angle 0.011111111111111112)))
   (* 0.011111111111111112 (* angle (* PI (* b_m b_m))))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = (b_m + a_m) * (((b_m - a_m) * ((double) M_PI)) * (angle * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b_m * b_m)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = (b_m + a_m) * (((b_m - a_m) * Math.PI) * (angle * 0.011111111111111112));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b_m * b_m)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+177:
		tmp = (b_m + a_m) * (((b_m - a_m) * math.pi) * (angle * 0.011111111111111112))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b_m * b_m)))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+177)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(Float64(b_m - a_m) * pi) * Float64(angle * 0.011111111111111112)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b_m * b_m))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+177)
		tmp = (b_m + a_m) * (((b_m - a_m) * pi) * (angle * 0.011111111111111112));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b_m * b_m)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+177], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e177

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower--.f64 62.8

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

   7. Applied rewrites 62.8%

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

   if 2e177 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 31.7

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

   5. Applied rewrites 31.7%

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 38.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(b - a\right) \cdot \pi\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.7% accurate, 11.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;a\_m \cdot \left(-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+177)
   (* a_m (* -0.011111111111111112 (* a_m (* angle PI))))
   (* (* angle PI) (* 0.011111111111111112 (* a_m a_m)))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * ((double) M_PI))));
	} else {
		tmp = (angle * ((double) M_PI)) * (0.011111111111111112 * (a_m * a_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * Math.PI)));
	} else {
		tmp = (angle * Math.PI) * (0.011111111111111112 * (a_m * a_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+177:
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * math.pi)))
	else:
		tmp = (angle * math.pi) * (0.011111111111111112 * (a_m * a_m))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+177)
		tmp = Float64(a_m * Float64(-0.011111111111111112 * Float64(a_m * Float64(angle * pi))));
	else
		tmp = Float64(Float64(angle * pi) * Float64(0.011111111111111112 * Float64(a_m * a_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+177)
		tmp = a_m * (-0.011111111111111112 * (a_m * (angle * pi)));
	else
		tmp = (angle * pi) * (0.011111111111111112 * (a_m * a_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+177], N[(a$95$m * N[(-0.011111111111111112 * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * Pi), $MachinePrecision] * N[(0.011111111111111112 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e177

   1. Initial program 52.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 30.3%

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

   10. Applied rewrites 38.3%

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

   if 2e177 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.4%

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 23.6%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 29.0%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(a\_m \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+177)
   (* -0.011111111111111112 (* a_m (* a_m (* angle PI))))
   (* (* angle PI) (* 0.011111111111111112 (* a_m a_m)))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = -0.011111111111111112 * (a_m * (a_m * (angle * ((double) M_PI))));
	} else {
		tmp = (angle * ((double) M_PI)) * (0.011111111111111112 * (a_m * a_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = -0.011111111111111112 * (a_m * (a_m * (angle * Math.PI)));
	} else {
		tmp = (angle * Math.PI) * (0.011111111111111112 * (a_m * a_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+177:
		tmp = -0.011111111111111112 * (a_m * (a_m * (angle * math.pi)))
	else:
		tmp = (angle * math.pi) * (0.011111111111111112 * (a_m * a_m))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+177)
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(a_m * Float64(angle * pi))));
	else
		tmp = Float64(Float64(angle * pi) * Float64(0.011111111111111112 * Float64(a_m * a_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+177)
		tmp = -0.011111111111111112 * (a_m * (a_m * (angle * pi)));
	else
		tmp = (angle * pi) * (0.011111111111111112 * (a_m * a_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+177], N[(-0.011111111111111112 * N[(a$95$m * N[(a$95$m * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * Pi), $MachinePrecision] * N[(0.011111111111111112 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e177

   1. Initial program 52.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 30.3%

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

   10. Applied rewrites 38.3%

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

   if 2e177 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.4%

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 23.6%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 29.0%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a\_m \cdot a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+177)
   (* -0.011111111111111112 (* (* angle PI) (* a_m a_m)))
   (* (* angle PI) (* 0.011111111111111112 (* a_m a_m)))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = -0.011111111111111112 * ((angle * ((double) M_PI)) * (a_m * a_m));
	} else {
		tmp = (angle * ((double) M_PI)) * (0.011111111111111112 * (a_m * a_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+177) {
		tmp = -0.011111111111111112 * ((angle * Math.PI) * (a_m * a_m));
	} else {
		tmp = (angle * Math.PI) * (0.011111111111111112 * (a_m * a_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	tmp = 0
	if (angle / 180.0) <= 2e+177:
		tmp = -0.011111111111111112 * ((angle * math.pi) * (a_m * a_m))
	else:
		tmp = (angle * math.pi) * (0.011111111111111112 * (a_m * a_m))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+177)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(angle * pi) * Float64(a_m * a_m)));
	else
		tmp = Float64(Float64(angle * pi) * Float64(0.011111111111111112 * Float64(a_m * a_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= 2e+177)
		tmp = -0.011111111111111112 * ((angle * pi) * (a_m * a_m));
	else
		tmp = (angle * pi) * (0.011111111111111112 * (a_m * a_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+177], N[(-0.011111111111111112 * N[(N[(angle * Pi), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle * Pi), $MachinePrecision] * N[(0.011111111111111112 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e177

   1. Initial program 52.4%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 30.3%

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

   if 2e177 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 27.4%

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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 23.6%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 29.0%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+177}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 20.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ \left(angle \cdot \pi\right) \cdot \left(0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle)
 :precision binary64
 (* (* angle PI) (* 0.011111111111111112 (* a_m a_m))))

C

b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle) {
	return (angle * ((double) M_PI)) * (0.011111111111111112 * (a_m * a_m));
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle) {
	return (angle * Math.PI) * (0.011111111111111112 * (a_m * a_m));
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle):
	return (angle * math.pi) * (0.011111111111111112 * (a_m * a_m))

Julia

b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle)
	return Float64(Float64(angle * pi) * Float64(0.011111111111111112 * Float64(a_m * a_m)))
end

MATLAB

b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle)
	tmp = (angle * pi) * (0.011111111111111112 * (a_m * a_m));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_] := N[(N[(angle * Pi), $MachinePrecision] * N[(0.011111111111111112 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.6%

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

4. Applied rewrites 6.1%

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

5. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-out N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. metadata-eval N/A

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

   10. *-rgt-identity N/A

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

   11. lower-*.f64 N/A

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

7. Applied rewrites 20.5%

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

8. Taylor expanded in angle around 0

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

10. Applied rewrites 19.4%

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

11. Final simplification 19.4%

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

Reproduce

herbie shell --seed 2024234 
(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)))))