ab-angle->ABCF B

Percentage Accurate: 54.1% → 63.8%

Time: 5.1s

Alternatives: 10

Speedup: 13.7×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% 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: 63.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.6 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\left(a\_m + b\right) \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(b - a\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle\_m \cdot \pi\right) \cdot \left(\left(\left(b - a\_m\right) \cdot \left(a\_m + b\right)\right) \cdot 0.011111111111111112\right)\right) \cdot \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.6e+19)
    (* (* (* (+ a_m b) PI) angle_m) (* (- b a_m) 0.011111111111111112))
    (*
     (* (* angle_m PI) (* (* (- b a_m) (+ a_m b)) 0.011111111111111112))
     (cos (* PI (/ angle_m 180.0)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e+19) {
		tmp = (((a_m + b) * ((double) M_PI)) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = ((angle_m * ((double) M_PI)) * (((b - a_m) * (a_m + b)) * 0.011111111111111112)) * cos((((double) M_PI) * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e+19) {
		tmp = (((a_m + b) * Math.PI) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = ((angle_m * Math.PI) * (((b - a_m) * (a_m + b)) * 0.011111111111111112)) * Math.cos((Math.PI * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 4.6e+19:
		tmp = (((a_m + b) * math.pi) * angle_m) * ((b - a_m) * 0.011111111111111112)
	else:
		tmp = ((angle_m * math.pi) * (((b - a_m) * (a_m + b)) * 0.011111111111111112)) * math.cos((math.pi * (angle_m / 180.0)))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.6e+19)
		tmp = Float64(Float64(Float64(Float64(a_m + b) * pi) * angle_m) * Float64(Float64(b - a_m) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(angle_m * pi) * Float64(Float64(Float64(b - a_m) * Float64(a_m + b)) * 0.011111111111111112)) * cos(Float64(pi * Float64(angle_m / 180.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 4.6e+19)
		tmp = (((a_m + b) * pi) * angle_m) * ((b - a_m) * 0.011111111111111112);
	else
		tmp = ((angle_m * pi) * (((b - a_m) * (a_m + b)) * 0.011111111111111112)) * cos((pi * (angle_m / 180.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.6e+19], N[(N[(N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(N[(N[(b - a$95$m), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.6e19

   1. Initial program 75.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 77.4

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

   5. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 95.4

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

   7. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 95.6%

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

   if 4.6e19 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 27.9

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

   5. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\left(\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 28.0

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

   7. Applied rewrites 28.0%

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

Alternative 2: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(a\_m \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(b - a\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\_m\right) \cdot \left(\left(\pi \cdot \left(a\_m + b\right)\right) \cdot \left(b - a\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot b\right) \cdot angle\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a_m 2.0)))))
   (*
    angle_s
    (if (<= t_0 (- INFINITY))
      (* (* (* a_m PI) angle_m) (* (- b a_m) 0.011111111111111112))
      (if (<= t_0 1e+304)
        (* (* 0.011111111111111112 angle_m) (* (* PI (+ a_m b)) (- b a_m)))
        (* (* (* (* PI b) angle_m) (- b a_m)) 0.011111111111111112))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a_m, 2.0));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((a_m * ((double) M_PI)) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else if (t_0 <= 1e+304) {
		tmp = (0.011111111111111112 * angle_m) * ((((double) M_PI) * (a_m + b)) * (b - a_m));
	} else {
		tmp = (((((double) M_PI) * b) * angle_m) * (b - a_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((a_m * Math.PI) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else if (t_0 <= 1e+304) {
		tmp = (0.011111111111111112 * angle_m) * ((Math.PI * (a_m + b)) * (b - a_m));
	} else {
		tmp = (((Math.PI * b) * angle_m) * (b - a_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((a_m * math.pi) * angle_m) * ((b - a_m) * 0.011111111111111112)
	elif t_0 <= 1e+304:
		tmp = (0.011111111111111112 * angle_m) * ((math.pi * (a_m + b)) * (b - a_m))
	else:
		tmp = (((math.pi * b) * angle_m) * (b - a_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(a_m * pi) * angle_m) * Float64(Float64(b - a_m) * 0.011111111111111112));
	elseif (t_0 <= 1e+304)
		tmp = Float64(Float64(0.011111111111111112 * angle_m) * Float64(Float64(pi * Float64(a_m + b)) * Float64(b - a_m)));
	else
		tmp = Float64(Float64(Float64(Float64(pi * b) * angle_m) * Float64(b - a_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((b ^ 2.0) - (a_m ^ 2.0));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((a_m * pi) * angle_m) * ((b - a_m) * 0.011111111111111112);
	elseif (t_0 <= 1e+304)
		tmp = (0.011111111111111112 * angle_m) * ((pi * (a_m + b)) * (b - a_m));
	else
		tmp = (((pi * b) * angle_m) * (b - a_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(a$95$m * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+304], N[(N[(0.011111111111111112 * angle$95$m), $MachinePrecision] * N[(N[(Pi * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * b), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -inf.0

   1. Initial program 52.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 52.0

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

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 74.7

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

   7. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 75.0%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 75.0%

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

   if -inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 9.9999999999999994e303

   1. Initial program 61.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 56.4

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

   5. Applied rewrites 56.4%

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

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

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      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. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

   7. Applied rewrites 56.4%

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

   if 9.9999999999999994e303 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 39.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 54.4

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

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 75.5

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 69.5

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

   10. Applied rewrites 69.5%

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

Alternative 3: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-166}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+299}:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot b\right) \cdot angle\_m\right) \cdot \left(b - a\_m\right)\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a_m 2.0)))))
   (*
    angle_s
    (if (<= t_0 -5e-166)
      (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m))
      (if (<= t_0 1e+299)
        (* (* (* PI angle_m) (* b b)) 0.011111111111111112)
        (* (* (* (* PI b) angle_m) (- b a_m)) 0.011111111111111112))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a_m, 2.0));
	double tmp;
	if (t_0 <= -5e-166) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	} else if (t_0 <= 1e+299) {
		tmp = ((((double) M_PI) * angle_m) * (b * b)) * 0.011111111111111112;
	} else {
		tmp = (((((double) M_PI) * b) * angle_m) * (b - a_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0));
	double tmp;
	if (t_0 <= -5e-166) {
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	} else if (t_0 <= 1e+299) {
		tmp = ((Math.PI * angle_m) * (b * b)) * 0.011111111111111112;
	} else {
		tmp = (((Math.PI * b) * angle_m) * (b - a_m)) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))
	tmp = 0
	if t_0 <= -5e-166:
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	elif t_0 <= 1e+299:
		tmp = ((math.pi * angle_m) * (b * b)) * 0.011111111111111112
	else:
		tmp = (((math.pi * b) * angle_m) * (b - a_m)) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	tmp = 0.0
	if (t_0 <= -5e-166)
		tmp = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m));
	elseif (t_0 <= 1e+299)
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(b * b)) * 0.011111111111111112);
	else
		tmp = Float64(Float64(Float64(Float64(pi * b) * angle_m) * Float64(b - a_m)) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = 2.0 * ((b ^ 2.0) - (a_m ^ 2.0));
	tmp = 0.0;
	if (t_0 <= -5e-166)
		tmp = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	elseif (t_0 <= 1e+299)
		tmp = ((pi * angle_m) * (b * b)) * 0.011111111111111112;
	else
		tmp = (((pi * b) * angle_m) * (b - a_m)) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$0, -5e-166], N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+299], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(N[(N[(Pi * b), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -5e-166

   1. Initial program 54.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   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 52.0

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

   8. Applied rewrites 52.0%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \pi\right)} \]
   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 52.2

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

   10. Applied rewrites 52.2%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(angle \cdot \pi\right) \]
   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 62.6

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

   12. Applied rewrites 62.6%

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

   if -5e-166 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 1.0000000000000001e299

   1. Initial program 64.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in a around 0

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

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

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if 1.0000000000000001e299 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 39.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 54.2

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

   5. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 69.2

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

   10. Applied rewrites 69.2%

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

Alternative 4: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ t_1 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(\pi \cdot angle\_m\right) \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))
        (t_1 (* 2.0 (- (pow b 2.0) (pow a_m 2.0)))))
   (*
    angle_s
    (if (<= t_1 -5e-166)
      t_0
      (if (<= t_1 INFINITY)
        (* (* (* PI angle_m) (* b b)) 0.011111111111111112)
        t_0)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a_m, 2.0));
	double tmp;
	if (t_1 <= -5e-166) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((((double) M_PI) * angle_m) * (b * b)) * 0.011111111111111112;
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	double t_1 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0));
	double tmp;
	if (t_1 <= -5e-166) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.PI * angle_m) * (b * b)) * 0.011111111111111112;
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	t_1 = 2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))
	tmp = 0
	if t_1 <= -5e-166:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((math.pi * angle_m) * (b * b)) * 0.011111111111111112
	else:
		tmp = t_0
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -5e-166)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(pi * angle_m) * Float64(b * b)) * 0.011111111111111112);
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	t_1 = 2.0 * ((b ^ 2.0) - (a_m ^ 2.0));
	tmp = 0.0;
	if (t_1 <= -5e-166)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((pi * angle_m) * (b * b)) * 0.011111111111111112;
	else
		tmp = t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, -5e-166], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], t$95$0]]), $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)))) < -5e-166 or +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 46.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

   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 50.3

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

   8. Applied rewrites 50.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \pi\right)} \]
   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 50.5

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

   10. Applied rewrites 50.5%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(angle \cdot \pi\right) \]
   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 60.6

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

   12. Applied rewrites 60.6%

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

   if -5e-166 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 60.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in a around 0

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

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

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 56.4

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

   8. Applied rewrites 56.4%

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

Alternative 5: 58.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(-0.011111111111111112 \cdot a\_m\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot a\_m\right)\\ t_1 := 2 \cdot \left({b}^{2} - {a\_m}^{2}\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\_m\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (* -0.011111111111111112 a_m) (* (* angle_m PI) a_m)))
        (t_1 (* 2.0 (- (pow b 2.0) (pow a_m 2.0)))))
   (*
    angle_s
    (if (<= t_1 -5e-166)
      t_0
      (if (<= t_1 INFINITY)
        (* (* (* PI (* b b)) angle_m) 0.011111111111111112)
        t_0)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (-0.011111111111111112 * a_m) * ((angle_m * ((double) M_PI)) * a_m);
	double t_1 = 2.0 * (pow(b, 2.0) - pow(a_m, 2.0));
	double tmp;
	if (t_1 <= -5e-166) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((((double) M_PI) * (b * b)) * angle_m) * 0.011111111111111112;
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (-0.011111111111111112 * a_m) * ((angle_m * Math.PI) * a_m);
	double t_1 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a_m, 2.0));
	double tmp;
	if (t_1 <= -5e-166) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.PI * (b * b)) * angle_m) * 0.011111111111111112;
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = (-0.011111111111111112 * a_m) * ((angle_m * math.pi) * a_m)
	t_1 = 2.0 * (math.pow(b, 2.0) - math.pow(a_m, 2.0))
	tmp = 0
	if t_1 <= -5e-166:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((math.pi * (b * b)) * angle_m) * 0.011111111111111112
	else:
		tmp = t_0
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(-0.011111111111111112 * a_m) * Float64(Float64(angle_m * pi) * a_m))
	t_1 = Float64(2.0 * Float64((b ^ 2.0) - (a_m ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -5e-166)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(pi * Float64(b * b)) * angle_m) * 0.011111111111111112);
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = (-0.011111111111111112 * a_m) * ((angle_m * pi) * a_m);
	t_1 = 2.0 * ((b ^ 2.0) - (a_m ^ 2.0));
	tmp = 0.0;
	if (t_1 <= -5e-166)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((pi * (b * b)) * angle_m) * 0.011111111111111112;
	else
		tmp = t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(N[(-0.011111111111111112 * a$95$m), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, -5e-166], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], t$95$0]]), $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)))) < -5e-166 or +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 46.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

   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 50.3

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

   8. Applied rewrites 50.3%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \pi\right)} \]
   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 50.5

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

   10. Applied rewrites 50.5%

       \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(angle \cdot \pi\right) \]
   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 60.6

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

   12. Applied rewrites 60.6%

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

   if -5e-166 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 60.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]
   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 56.3

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

   8. Applied rewrites 56.3%

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

Alternative 6: 64.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.6 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\left(a\_m + b\right) \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(b - a\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\pi \cdot angle\_m\right) \cdot \left(\left(b + a\_m\right) \cdot \left(b - a\_m\right)\right)\right) \cdot 0.011111111111111112\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.6e+19)
    (* (* (* (+ a_m b) PI) angle_m) (* (- b a_m) 0.011111111111111112))
    (*
     (* (* (* PI angle_m) (* (+ b a_m) (- b a_m))) 0.011111111111111112)
     (cos (* PI (* 0.005555555555555556 angle_m)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e+19) {
		tmp = (((a_m + b) * ((double) M_PI)) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = (((((double) M_PI) * angle_m) * ((b + a_m) * (b - a_m))) * 0.011111111111111112) * cos((((double) M_PI) * (0.005555555555555556 * angle_m)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 4.6e+19) {
		tmp = (((a_m + b) * Math.PI) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = (((Math.PI * angle_m) * ((b + a_m) * (b - a_m))) * 0.011111111111111112) * Math.cos((Math.PI * (0.005555555555555556 * angle_m)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 4.6e+19:
		tmp = (((a_m + b) * math.pi) * angle_m) * ((b - a_m) * 0.011111111111111112)
	else:
		tmp = (((math.pi * angle_m) * ((b + a_m) * (b - a_m))) * 0.011111111111111112) * math.cos((math.pi * (0.005555555555555556 * angle_m)))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 4.6e+19)
		tmp = Float64(Float64(Float64(Float64(a_m + b) * pi) * angle_m) * Float64(Float64(b - a_m) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(Float64(pi * angle_m) * Float64(Float64(b + a_m) * Float64(b - a_m))) * 0.011111111111111112) * cos(Float64(pi * Float64(0.005555555555555556 * angle_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 4.6e+19)
		tmp = (((a_m + b) * pi) * angle_m) * ((b - a_m) * 0.011111111111111112);
	else
		tmp = (((pi * angle_m) * ((b + a_m) * (b - a_m))) * 0.011111111111111112) * cos((pi * (0.005555555555555556 * angle_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.6e+19], N[(N[(N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(b + a$95$m), $MachinePrecision] * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.6e19

   1. Initial program 75.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 77.4

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

   5. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 95.4

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

   7. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 95.6%

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

   if 4.6e19 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \frac{1}{90}\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(\frac{1}{180} \cdot angle\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 28.4

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)\right) \]

   8. Applied rewrites 28.4%

      \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.4 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\left(a\_m + b\right) \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(b - a\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.4e+20)
    (* (* (* (+ a_m b) PI) angle_m) (* (- b a_m) 0.011111111111111112))
    (*
     (* (* -0.011111111111111112 (* a_m a_m)) (* angle_m PI))
     (cos (* PI (/ angle_m 180.0)))))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.4e+20) {
		tmp = (((a_m + b) * ((double) M_PI)) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = ((-0.011111111111111112 * (a_m * a_m)) * (angle_m * ((double) M_PI))) * cos((((double) M_PI) * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 1.4e+20) {
		tmp = (((a_m + b) * Math.PI) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = ((-0.011111111111111112 * (a_m * a_m)) * (angle_m * Math.PI)) * Math.cos((Math.PI * (angle_m / 180.0)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 1.4e+20:
		tmp = (((a_m + b) * math.pi) * angle_m) * ((b - a_m) * 0.011111111111111112)
	else:
		tmp = ((-0.011111111111111112 * (a_m * a_m)) * (angle_m * math.pi)) * math.cos((math.pi * (angle_m / 180.0)))
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 1.4e+20)
		tmp = Float64(Float64(Float64(Float64(a_m + b) * pi) * angle_m) * Float64(Float64(b - a_m) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(-0.011111111111111112 * Float64(a_m * a_m)) * Float64(angle_m * pi)) * cos(Float64(pi * Float64(angle_m / 180.0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.4e+20)
		tmp = (((a_m + b) * pi) * angle_m) * ((b - a_m) * 0.011111111111111112);
	else
		tmp = ((-0.011111111111111112 * (a_m * a_m)) * (angle_m * pi)) * cos((pi * (angle_m / 180.0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.4e+20], N[(N[(N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.011111111111111112 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.4e20

   1. Initial program 74.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 77.3

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

   5. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 95.3

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

   7. Applied rewrites 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 95.5%

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

   if 1.4e20 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 22.5

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

   8. Applied rewrites 22.5%

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

Alternative 8: 62.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.5 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\left(a\_m + b\right) \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(b - a\_m\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot \pi\right) \cdot \left(a\_m + b\right)\right) \cdot b\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2.5e+61)
    (* (* (* (+ a_m b) PI) angle_m) (* (- b a_m) 0.011111111111111112))
    (* (* (* (* angle_m PI) (+ a_m b)) b) 0.011111111111111112))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.5e+61) {
		tmp = (((a_m + b) * ((double) M_PI)) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = (((angle_m * ((double) M_PI)) * (a_m + b)) * b) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double tmp;
	if (angle_m <= 2.5e+61) {
		tmp = (((a_m + b) * Math.PI) * angle_m) * ((b - a_m) * 0.011111111111111112);
	} else {
		tmp = (((angle_m * Math.PI) * (a_m + b)) * b) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	tmp = 0
	if angle_m <= 2.5e+61:
		tmp = (((a_m + b) * math.pi) * angle_m) * ((b - a_m) * 0.011111111111111112)
	else:
		tmp = (((angle_m * math.pi) * (a_m + b)) * b) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	tmp = 0.0
	if (angle_m <= 2.5e+61)
		tmp = Float64(Float64(Float64(Float64(a_m + b) * pi) * angle_m) * Float64(Float64(b - a_m) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(Float64(angle_m * pi) * Float64(a_m + b)) * b) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 2.5e+61)
		tmp = (((a_m + b) * pi) * angle_m) * ((b - a_m) * 0.011111111111111112);
	else
		tmp = (((angle_m * pi) * (a_m + b)) * b) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2.5e+61], N[(N[(N[(N[(a$95$m + b), $MachinePrecision] * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(b - a$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.50000000000000009e61

   1. Initial program 69.9%

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

   3. 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 71.9

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

   5. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 87.7

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

   7. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 87.9%

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

   if 2.50000000000000009e61 < angle

   1. Initial program 30.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 29.8

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

   5. Applied rewrites 29.8%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. 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 25.6

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 25.4%

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

Alternative 9: 62.7% accurate, 13.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(angle\_m \cdot \pi\right) \cdot \left(a\_m + b\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.5 \cdot 10^{+61}:\\ \;\;\;\;\left(t\_0 \cdot \left(b - a\_m\right)\right) \cdot 0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot b\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b angle_m)
 :precision binary64
 (let* ((t_0 (* (* angle_m PI) (+ a_m b))))
   (*
    angle_s
    (if (<= angle_m 2.5e+61)
      (* (* t_0 (- b a_m)) 0.011111111111111112)
      (* (* t_0 b) 0.011111111111111112)))))

C

a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (angle_m * ((double) M_PI)) * (a_m + b);
	double tmp;
	if (angle_m <= 2.5e+61) {
		tmp = (t_0 * (b - a_m)) * 0.011111111111111112;
	} else {
		tmp = (t_0 * b) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b, double angle_m) {
	double t_0 = (angle_m * Math.PI) * (a_m + b);
	double tmp;
	if (angle_m <= 2.5e+61) {
		tmp = (t_0 * (b - a_m)) * 0.011111111111111112;
	} else {
		tmp = (t_0 * b) * 0.011111111111111112;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b, angle_m):
	t_0 = (angle_m * math.pi) * (a_m + b)
	tmp = 0
	if angle_m <= 2.5e+61:
		tmp = (t_0 * (b - a_m)) * 0.011111111111111112
	else:
		tmp = (t_0 * b) * 0.011111111111111112
	return angle_s * tmp

Julia

a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b, angle_m)
	t_0 = Float64(Float64(angle_m * pi) * Float64(a_m + b))
	tmp = 0.0
	if (angle_m <= 2.5e+61)
		tmp = Float64(Float64(t_0 * Float64(b - a_m)) * 0.011111111111111112);
	else
		tmp = Float64(Float64(t_0 * b) * 0.011111111111111112);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b, angle_m)
	t_0 = (angle_m * pi) * (a_m + b);
	tmp = 0.0;
	if (angle_m <= 2.5e+61)
		tmp = (t_0 * (b - a_m)) * 0.011111111111111112;
	else
		tmp = (t_0 * b) * 0.011111111111111112;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $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_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a$95$m + b), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 2.5e+61], N[(N[(t$95$0 * N[(b - a$95$m), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision], N[(N[(t$95$0 * b), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.50000000000000009e61

   1. Initial program 69.9%

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

   3. 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 71.9

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

   5. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   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 87.7

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

   7. Applied rewrites 87.7%

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

   if 2.50000000000000009e61 < angle

   1. Initial program 30.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 29.8

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

   5. Applied rewrites 29.8%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot 0.011111111111111112} \]
   6. 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 25.6

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

   7. Applied rewrites 25.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 25.4%

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

Alternative 10: 39.2% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

a_m = N[Abs[a], $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_, 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 54.1%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
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 55.1

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

5. Applied rewrites 55.1%

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

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 35.7

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

8. Applied rewrites 35.7%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \pi\right)} \]
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 35.8

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

10. Applied rewrites 35.8%

    \[\leadsto \left(\left(-0.011111111111111112 \cdot a\right) \cdot a\right) \cdot \left(angle \cdot \pi\right) \]
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 39.2

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

12. Applied rewrites 39.2%

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

Reproduce

herbie shell --seed 2025088 
(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)))))