ab-angle->ABCF B

Percentage Accurate: 53.7% → 67.0%

Time: 6.4s

Alternatives: 10

Speedup: 13.7×

• 32.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot 0.011111111111111112\\ \mathbf{elif}\;angle\_m \leq 3.5 \cdot 10^{+230}:\\ \;\;\;\;\sin \left(\left(2 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot 0.005555555555555556\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(a\_m + b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2e-36)
    (* (* (* (* angle_m PI) (+ a_m b_m)) (- b_m a_m)) 0.011111111111111112)
    (if (<= angle_m 3.5e+230)
      (*
       (sin (* (* 2.0 (* angle_m PI)) 0.005555555555555556))
       (* (- b_m a_m) (+ a_m b_m)))
      (* (* (* PI angle_m) (* (+ b_m a_m) b_m)) 0.011111111111111112)))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2e-36) {
		tmp = (((angle_m * ((double) M_PI)) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112;
	} else if (angle_m <= 3.5e+230) {
		tmp = sin(((2.0 * (angle_m * ((double) M_PI))) * 0.005555555555555556)) * ((b_m - a_m) * (a_m + b_m));
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2e-36) {
		tmp = (((angle_m * Math.PI) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112;
	} else if (angle_m <= 3.5e+230) {
		tmp = Math.sin(((2.0 * (angle_m * Math.PI)) * 0.005555555555555556)) * ((b_m - a_m) * (a_m + b_m));
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 2e-36:
		tmp = (((angle_m * math.pi) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112
	elif angle_m <= 3.5e+230:
		tmp = math.sin(((2.0 * (angle_m * math.pi)) * 0.005555555555555556)) * ((b_m - a_m) * (a_m + b_m))
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 2e-36)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * Float64(a_m + b_m)) * Float64(b_m - a_m)) * 0.011111111111111112);
	elseif (angle_m <= 3.5e+230)
		tmp = Float64(sin(Float64(Float64(2.0 * Float64(angle_m * pi)) * 0.005555555555555556)) * Float64(Float64(b_m - a_m) * Float64(a_m + b_m)));
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a_m) * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 2e-36)
		tmp = (((angle_m * pi) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112;
	elseif (angle_m <= 3.5e+230)
		tmp = sin(((2.0 * (angle_m * pi)) * 0.005555555555555556)) * ((b_m - a_m) * (a_m + b_m));
	else
		tmp = ((pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2e-36], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], If[LessEqual[angle$95$m, 3.5e+230], N[(N[Sin[N[(N[(2.0 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 1.9999999999999999e-36

   1. Initial program 57.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if 1.9999999999999999e-36 < angle < 3.5e230

   1. Initial program 43.8%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 50.7%

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

   if 3.5e230 < angle

   1. Initial program 30.7%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 19.0

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 30.9%

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

Alternative 2: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.5 \cdot 10^{+230}:\\ \;\;\;\;\left(\left(\left(a\_m + b\_m\right) \cdot \sin t\_0\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \cos t\_0\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.005555555555555556)))
   (*
    angle_s
    (if (<= angle_m 3.5e+230)
      (* (* (* (+ a_m b_m) (sin t_0)) (* (- b_m a_m) (cos t_0))) 2.0)
      (* (* (* PI angle_m) (* (+ b_m a_m) b_m)) 0.011111111111111112)))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if (angle_m <= 3.5e+230) {
		tmp = (((a_m + b_m) * sin(t_0)) * ((b_m - a_m) * cos(t_0))) * 2.0;
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m * Math.PI) * 0.005555555555555556;
	double tmp;
	if (angle_m <= 3.5e+230) {
		tmp = (((a_m + b_m) * Math.sin(t_0)) * ((b_m - a_m) * Math.cos(t_0))) * 2.0;
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (angle_m * math.pi) * 0.005555555555555556
	tmp = 0
	if angle_m <= 3.5e+230:
		tmp = (((a_m + b_m) * math.sin(t_0)) * ((b_m - a_m) * math.cos(t_0))) * 2.0
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	tmp = 0.0
	if (angle_m <= 3.5e+230)
		tmp = Float64(Float64(Float64(Float64(a_m + b_m) * sin(t_0)) * Float64(Float64(b_m - a_m) * cos(t_0))) * 2.0);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a_m) * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (angle_m * pi) * 0.005555555555555556;
	tmp = 0.0;
	if (angle_m <= 3.5e+230)
		tmp = (((a_m + b_m) * sin(t_0)) * ((b_m - a_m) * cos(t_0))) * 2.0;
	else
		tmp = ((pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 3.5e+230], N[(N[(N[(N[(a$95$m + b$95$m), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.5e230

   1. Initial program 53.6%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 69.6

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

   9. Applied rewrites 69.6%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-PI.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-cos.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 N/A

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

       13. lift-*.f64 N/A

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

   11. Applied rewrites 70.5%

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

   if 3.5e230 < angle

   1. Initial program 30.7%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 19.0

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 30.9%

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

Alternative 3: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.5 \cdot 10^{+230}:\\ \;\;\;\;\left(\left(\left(\sin t\_0 \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \cos t\_0\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) 0.005555555555555556)))
   (*
    angle_s
    (if (<= angle_m 3.5e+230)
      (* (* (* (* (sin t_0) (+ a_m b_m)) (- b_m a_m)) (cos t_0)) 2.0)
      (* (* (* PI angle_m) (* (+ b_m a_m) b_m)) 0.011111111111111112)))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * 0.005555555555555556;
	double tmp;
	if (angle_m <= 3.5e+230) {
		tmp = (((sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * cos(t_0)) * 2.0;
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (angle_m * Math.PI) * 0.005555555555555556;
	double tmp;
	if (angle_m <= 3.5e+230) {
		tmp = (((Math.sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * Math.cos(t_0)) * 2.0;
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (angle_m * math.pi) * 0.005555555555555556
	tmp = 0
	if angle_m <= 3.5e+230:
		tmp = (((math.sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * math.cos(t_0)) * 2.0
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * 0.005555555555555556)
	tmp = 0.0
	if (angle_m <= 3.5e+230)
		tmp = Float64(Float64(Float64(Float64(sin(t_0) * Float64(a_m + b_m)) * Float64(b_m - a_m)) * cos(t_0)) * 2.0);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a_m) * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (angle_m * pi) * 0.005555555555555556;
	tmp = 0.0;
	if (angle_m <= 3.5e+230)
		tmp = (((sin(t_0) * (a_m + b_m)) * (b_m - a_m)) * cos(t_0)) * 2.0;
	else
		tmp = ((pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 3.5e+230], N[(N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.5e230

   1. Initial program 53.6%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 70.4%

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

   if 3.5e230 < angle

   1. Initial program 30.7%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 19.0

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 30.9%

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

Alternative 4: 66.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.5 \cdot 10^{+196}:\\ \;\;\;\;\left(\left(\left(\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right) \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2.5e+196)
    (*
     (*
      (*
       (* (sin (* (* angle_m PI) 0.005555555555555556)) (+ a_m b_m))
       (- b_m a_m))
      (cos (* (* 0.005555555555555556 angle_m) PI)))
     2.0)
    (* (* (* PI angle_m) (* (+ b_m a_m) b_m)) 0.011111111111111112))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2.5e+196) {
		tmp = (((sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)) * (a_m + b_m)) * (b_m - a_m)) * cos(((0.005555555555555556 * angle_m) * ((double) M_PI)))) * 2.0;
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2.5e+196) {
		tmp = (((Math.sin(((angle_m * Math.PI) * 0.005555555555555556)) * (a_m + b_m)) * (b_m - a_m)) * Math.cos(((0.005555555555555556 * angle_m) * Math.PI))) * 2.0;
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 2.5e+196:
		tmp = (((math.sin(((angle_m * math.pi) * 0.005555555555555556)) * (a_m + b_m)) * (b_m - a_m)) * math.cos(((0.005555555555555556 * angle_m) * math.pi))) * 2.0
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 2.5e+196)
		tmp = Float64(Float64(Float64(Float64(sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) * Float64(a_m + b_m)) * Float64(b_m - a_m)) * cos(Float64(Float64(0.005555555555555556 * angle_m) * pi))) * 2.0);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a_m) * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 2.5e+196)
		tmp = (((sin(((angle_m * pi) * 0.005555555555555556)) * (a_m + b_m)) * (b_m - a_m)) * cos(((0.005555555555555556 * angle_m) * pi))) * 2.0;
	else
		tmp = ((pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2.5e+196], N[(N[(N[(N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.4999999999999999e196

   1. Initial program 54.7%

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

   3. Taylor expanded in angle around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 71.6

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

   9. Applied rewrites 71.6%

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

   if 2.4999999999999999e196 < angle

   1. Initial program 31.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 18.6

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 29.4%

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

Alternative 5: 57.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a\_m}^{2}\right) \leq -3 \cdot 10^{-193}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(b\_m \cdot \left(b\_m - a\_m\right)\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a_m 2.0))) -3e-193)
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
    (* (* (* PI angle_m) (* b_m (- b_m a_m))) 0.011111111111111112))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a_m, 2.0))) <= -3e-193) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else {
		tmp = ((((double) M_PI) * angle_m) * (b_m * (b_m - a_m))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0))) <= -3e-193) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	} else {
		tmp = ((Math.PI * angle_m) * (b_m * (b_m - a_m))) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a_m, 2.0))) <= -3e-193:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	else:
		tmp = ((math.pi * angle_m) * (b_m * (b_m - a_m))) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a_m ^ 2.0))) <= -3e-193)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(b_m * Float64(b_m - a_m))) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a_m ^ 2.0))) <= -3e-193)
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	else
		tmp = ((pi * angle_m) * (b_m * (b_m - a_m))) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3e-193], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b$95$m * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -2.9999999999999999e-193

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 43.8

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

   5. Applied rewrites 43.8%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 43.2

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

   8. Applied rewrites 43.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.2

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

   10. Applied rewrites 43.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 49.9

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

   12. Applied rewrites 49.9%

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

   if -2.9999999999999999e-193 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 49.8%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 54.0%

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

Alternative 6: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a\_m}^{2}\right) \leq -3 \cdot 10^{-193}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (* 2.0 (- (pow b_m 2.0) (pow a_m 2.0))) -3e-193)
    (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
    (* (* (* PI (* b_m b_m)) angle_m) 0.011111111111111112))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a_m, 2.0))) <= -3e-193) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else {
		tmp = ((((double) M_PI) * (b_m * b_m)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0))) <= -3e-193) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	} else {
		tmp = ((Math.PI * (b_m * b_m)) * angle_m) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a_m, 2.0))) <= -3e-193:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	else:
		tmp = ((math.pi * (b_m * b_m)) * angle_m) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a_m ^ 2.0))) <= -3e-193)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	else
		tmp = Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle_m) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a_m ^ 2.0))) <= -3e-193)
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	else
		tmp = ((pi * (b_m * b_m)) * angle_m) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -3e-193], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -2.9999999999999999e-193

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 43.8

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

   5. Applied rewrites 43.8%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 43.2

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

   8. Applied rewrites 43.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 43.2

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

   10. Applied rewrites 43.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-PI.f64 49.9

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

   12. Applied rewrites 49.9%

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

   if -2.9999999999999999e-193 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 49.8%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.8

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

   8. Applied rewrites 52.8%

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

Alternative 7: 63.0% accurate, 13.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9.8 \cdot 10^{+194}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9.8e+194)
    (* (* (* (* angle_m PI) (+ a_m b_m)) (- b_m a_m)) 0.011111111111111112)
    (* (* (* PI angle_m) (* (+ b_m a_m) b_m)) 0.011111111111111112))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9.8e+194) {
		tmp = (((angle_m * ((double) M_PI)) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112;
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9.8e+194) {
		tmp = (((angle_m * Math.PI) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112;
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 9.8e+194:
		tmp = (((angle_m * math.pi) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 9.8e+194)
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * Float64(a_m + b_m)) * Float64(b_m - a_m)) * 0.011111111111111112);
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a_m) * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 9.8e+194)
		tmp = (((angle_m * pi) * (a_m + b_m)) * (b_m - a_m)) * 0.011111111111111112;
	else
		tmp = ((pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9.8e+194], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.80000000000000053e194

   1. Initial program 54.5%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lift--.f64 66.6

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

   7. Applied rewrites 66.6%

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

   if 9.80000000000000053e194 < angle

   1. Initial program 33.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 28.4%

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

Alternative 8: 53.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9.8 \cdot 10^{+194}:\\ \;\;\;\;\left(angle\_m \cdot \pi\right) \cdot \left(\left(\left(b\_m - a\_m\right) \cdot \left(a\_m + b\_m\right)\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9.8e+194)
    (* (* angle_m PI) (* (* (- b_m a_m) (+ a_m b_m)) 0.011111111111111112))
    (* (* (* PI angle_m) (* (+ b_m a_m) b_m)) 0.011111111111111112))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9.8e+194) {
		tmp = (angle_m * ((double) M_PI)) * (((b_m - a_m) * (a_m + b_m)) * 0.011111111111111112);
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9.8e+194) {
		tmp = (angle_m * Math.PI) * (((b_m - a_m) * (a_m + b_m)) * 0.011111111111111112);
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 9.8e+194:
		tmp = (angle_m * math.pi) * (((b_m - a_m) * (a_m + b_m)) * 0.011111111111111112)
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 9.8e+194)
		tmp = Float64(Float64(angle_m * pi) * Float64(Float64(Float64(b_m - a_m) * Float64(a_m + b_m)) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a_m) * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 9.8e+194)
		tmp = (angle_m * pi) * (((b_m - a_m) * (a_m + b_m)) * 0.011111111111111112);
	else
		tmp = ((pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9.8e+194], N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.80000000000000053e194

   1. Initial program 54.5%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 55.8%

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

   if 9.80000000000000053e194 < angle

   1. Initial program 33.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 28.4%

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

Alternative 9: 53.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9.8 \cdot 10^{+194}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\left(\pi \cdot \left(a\_m + b\_m\right)\right) \cdot \left(b\_m - a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b\_m + a\_m\right) \cdot b\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 9.8e+194)
    (* (* 0.011111111111111112 angle_m) (* (* PI (+ a_m b_m)) (- b_m a_m)))
    (* (* (* PI angle_m) (* (+ b_m a_m) b_m)) 0.011111111111111112))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9.8e+194) {
		tmp = (0.011111111111111112 * angle_m) * ((((double) M_PI) * (a_m + b_m)) * (b_m - a_m));
	} else {
		tmp = ((((double) M_PI) * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 9.8e+194) {
		tmp = (0.011111111111111112 * angle_m) * ((Math.PI * (a_m + b_m)) * (b_m - a_m));
	} else {
		tmp = ((Math.PI * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 9.8e+194:
		tmp = (0.011111111111111112 * angle_m) * ((math.pi * (a_m + b_m)) * (b_m - a_m))
	else:
		tmp = ((math.pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 9.8e+194)
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(Float64(pi * Float64(a_m + b_m)) * Float64(b_m - a_m)));
	else
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b_m + a_m) * b_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 9.8e+194)
		tmp = (0.011111111111111112 * angle_m) * ((pi * (a_m + b_m)) * (b_m - a_m));
	else
		tmp = ((pi * angle_m) * ((b_m + a_m) * b_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9.8e+194], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(N[(Pi * N[(a$95$m + b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.80000000000000053e194

   1. Initial program 54.5%

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

   3. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. pow2 N/A

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

      16. pow2 N/A

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

   7. Applied rewrites 55.0%

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

   if 9.80000000000000053e194 < angle

   1. Initial program 33.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 28.4%

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

Alternative 10: 37.9% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower--.f64 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 29.2

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

8. Applied rewrites 29.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 29.2

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

10. Applied rewrites 29.2%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lift-PI.f64 31.1

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

12. Applied rewrites 31.1%

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

Reproduce

herbie shell --seed 2025084 
(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)))))