ab-angle->ABCF B

Percentage Accurate: 54.3% → 66.6%

Time: 16.8s

Alternatives: 15

Speedup: 10.3×

• 33.1% 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: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+29)
      (* (+ b a) (* (- b a) (sin (* PI (* angle_m 0.011111111111111112)))))
      (if (<= (/ angle_m 180.0) 5e+203)
        (* (sin (* 0.011111111111111112 (* angle_m PI))) t_0)
        (* angle_m (* PI (* 0.011111111111111112 t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 5e+29) {
		tmp = (b + a) * ((b - a) * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	} else if ((angle_m / 180.0) <= 5e+203) {
		tmp = sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * t_0;
	} else {
		tmp = angle_m * (((double) M_PI) * (0.011111111111111112 * t_0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 5e+29) {
		tmp = (b + a) * ((b - a) * Math.sin((Math.PI * (angle_m * 0.011111111111111112))));
	} else if ((angle_m / 180.0) <= 5e+203) {
		tmp = Math.sin((0.011111111111111112 * (angle_m * Math.PI))) * t_0;
	} else {
		tmp = angle_m * (Math.PI * (0.011111111111111112 * t_0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b + a) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 5e+29:
		tmp = (b + a) * ((b - a) * math.sin((math.pi * (angle_m * 0.011111111111111112))))
	elif (angle_m / 180.0) <= 5e+203:
		tmp = math.sin((0.011111111111111112 * (angle_m * math.pi))) * t_0
	else:
		tmp = angle_m * (math.pi * (0.011111111111111112 * t_0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+29)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	elseif (Float64(angle_m / 180.0) <= 5e+203)
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * t_0);
	else
		tmp = Float64(angle_m * Float64(pi * Float64(0.011111111111111112 * t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (b + a) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+29)
		tmp = (b + a) * ((b - a) * sin((pi * (angle_m * 0.011111111111111112))));
	elseif ((angle_m / 180.0) <= 5e+203)
		tmp = sin((0.011111111111111112 * (angle_m * pi))) * t_0;
	else
		tmp = angle_m * (pi * (0.011111111111111112 * t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+29], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+203], N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(angle$95$m * N[(Pi * N[(0.011111111111111112 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000001e29

   1. Initial program 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.0

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

   6. Applied rewrites 77.0%

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

   if 5.0000000000000001e29 < (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999994e203

   1. Initial program 23.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

   4. Applied rewrites 38.6%

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

   if 4.99999999999999994e203 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 20.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ t_1 := {b}^{2} - {a}^{2}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-297}:\\ \;\;\;\;\left(b + a\right) \cdot \left(t\_0 \cdot \left(-a\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* angle_m PI))))
        (t_1 (- (pow b 2.0) (pow a 2.0))))
   (*
    angle_s
    (if (<= t_1 -4e-297)
      (* (+ b a) (* t_0 (- a)))
      (if (<= t_1 INFINITY)
        (* (+ b a) (* b t_0))
        (*
         (+ b a)
         (*
          (- b a)
          (*
           angle_m
           (fma
            (* -2.2862368541380886e-7 (* angle_m angle_m))
            (* PI (* PI PI))
            (* PI 0.011111111111111112))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = sin((0.011111111111111112 * (angle_m * ((double) M_PI))));
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_1 <= -4e-297) {
		tmp = (b + a) * (t_0 * -a);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (b + a) * (b * t_0);
	} else {
		tmp = (b + a) * ((b - a) * (angle_m * fma((-2.2862368541380886e-7 * (angle_m * angle_m)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (((double) M_PI) * 0.011111111111111112))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_1 <= -4e-297)
		tmp = Float64(Float64(b + a) * Float64(t_0 * Float64(-a)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(b + a) * Float64(b * t_0));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(angle_m * fma(Float64(-2.2862368541380886e-7 * Float64(angle_m * angle_m)), Float64(pi * Float64(pi * pi)), Float64(pi * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[t$95$1, -4e-297], N[(N[(b + a), $MachinePrecision] * N[(t$95$0 * (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(b + a), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(-2.2862368541380886e-7 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.00000000000000016e-297

   1. Initial program 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 65.3

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

   7. Applied rewrites 65.3%

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

   if -4.00000000000000016e-297 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < +inf.0

   1. Initial program 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 65.2

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

   7. Applied rewrites 65.2%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 77.6

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

   6. Applied rewrites 77.6%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 88.7

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

   9. Applied rewrites 88.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -4 \cdot 10^{-297}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-a\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq \infty:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b 2.0) 2e-65)
    (* (+ b a) (* (sin (* angle_m (* PI 0.011111111111111112))) (- a)))
    (* (+ b a) (* (- b a) (* 0.011111111111111112 (* angle_m PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (pow(b, 2.0) <= 2e-65) {
		tmp = (b + a) * (sin((angle_m * (((double) M_PI) * 0.011111111111111112))) * -a);
	} else {
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (Math.pow(b, 2.0) <= 2e-65) {
		tmp = (b + a) * (Math.sin((angle_m * (Math.PI * 0.011111111111111112))) * -a);
	} else {
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * Math.PI)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if math.pow(b, 2.0) <= 2e-65:
		tmp = (b + a) * (math.sin((angle_m * (math.pi * 0.011111111111111112))) * -a)
	else:
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * math.pi)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if ((b ^ 2.0) <= 2e-65)
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(angle_m * Float64(pi * 0.011111111111111112))) * Float64(-a)));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle_m * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((b ^ 2.0) <= 2e-65)
		tmp = (b + a) * (sin((angle_m * (pi * 0.011111111111111112))) * -a);
	else
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 2e-65], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 1.99999999999999985e-65

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.5

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

   9. Applied rewrites 67.5%

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

   if 1.99999999999999985e-65 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 63.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 63.1

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

   7. Applied rewrites 63.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 4 \cdot 10^{-217}:\\ \;\;\;\;a \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(a \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) 4e-217)
    (* a (* (* angle_m PI) (* a -0.011111111111111112)))
    (* b (* 0.011111111111111112 (* b (* angle_m PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= 4e-217) {
		tmp = a * ((angle_m * ((double) M_PI)) * (a * -0.011111111111111112));
	} else {
		tmp = b * (0.011111111111111112 * (b * (angle_m * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= 4e-217) {
		tmp = a * ((angle_m * Math.PI) * (a * -0.011111111111111112));
	} else {
		tmp = b * (0.011111111111111112 * (b * (angle_m * Math.PI)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= 4e-217:
		tmp = a * ((angle_m * math.pi) * (a * -0.011111111111111112))
	else:
		tmp = b * (0.011111111111111112 * (b * (angle_m * math.pi)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= 4e-217)
		tmp = Float64(a * Float64(Float64(angle_m * pi) * Float64(a * -0.011111111111111112)));
	else
		tmp = Float64(b * Float64(0.011111111111111112 * Float64(b * Float64(angle_m * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= 4e-217)
		tmp = a * ((angle_m * pi) * (a * -0.011111111111111112));
	else
		tmp = b * (0.011111111111111112 * (b * (angle_m * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 4e-217], N[(a * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(0.011111111111111112 * N[(b * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.00000000000000033e-217

   1. Initial program 59.6%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.3%

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

   10. Applied rewrites 63.7%

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

   if 4.00000000000000033e-217 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 39.8%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 9.5%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 57.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 4 \cdot 10^{-217}:\\ \;\;\;\;a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 4 \cdot 10^{-217}:\\ \;\;\;\;\left(a \cdot \left(angle\_m \cdot a\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) 4e-217)
    (* (* a (* angle_m a)) (* PI -0.011111111111111112))
    (* b (* 0.011111111111111112 (* b (* angle_m PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= 4e-217) {
		tmp = (a * (angle_m * a)) * (((double) M_PI) * -0.011111111111111112);
	} else {
		tmp = b * (0.011111111111111112 * (b * (angle_m * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= 4e-217) {
		tmp = (a * (angle_m * a)) * (Math.PI * -0.011111111111111112);
	} else {
		tmp = b * (0.011111111111111112 * (b * (angle_m * Math.PI)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= 4e-217:
		tmp = (a * (angle_m * a)) * (math.pi * -0.011111111111111112)
	else:
		tmp = b * (0.011111111111111112 * (b * (angle_m * math.pi)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= 4e-217)
		tmp = Float64(Float64(a * Float64(angle_m * a)) * Float64(pi * -0.011111111111111112));
	else
		tmp = Float64(b * Float64(0.011111111111111112 * Float64(b * Float64(angle_m * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= 4e-217)
		tmp = (a * (angle_m * a)) * (pi * -0.011111111111111112);
	else
		tmp = b * (0.011111111111111112 * (b * (angle_m * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 4e-217], N[(N[(a * N[(angle$95$m * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(b * N[(0.011111111111111112 * N[(b * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.00000000000000033e-217

   1. Initial program 59.6%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 58.3%

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

   13. Applied rewrites 63.7%

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

   if 4.00000000000000033e-217 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 39.8%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 9.5%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 57.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 4 \cdot 10^{-217}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 4 \cdot 10^{-217}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) 4e-217)
    (* a (* -0.011111111111111112 (* a (* angle_m PI))))
    (* b (* 0.011111111111111112 (* b (* angle_m PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= 4e-217) {
		tmp = a * (-0.011111111111111112 * (a * (angle_m * ((double) M_PI))));
	} else {
		tmp = b * (0.011111111111111112 * (b * (angle_m * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= 4e-217) {
		tmp = a * (-0.011111111111111112 * (a * (angle_m * Math.PI)));
	} else {
		tmp = b * (0.011111111111111112 * (b * (angle_m * Math.PI)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= 4e-217:
		tmp = a * (-0.011111111111111112 * (a * (angle_m * math.pi)))
	else:
		tmp = b * (0.011111111111111112 * (b * (angle_m * math.pi)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= 4e-217)
		tmp = Float64(a * Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))));
	else
		tmp = Float64(b * Float64(0.011111111111111112 * Float64(b * Float64(angle_m * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= 4e-217)
		tmp = a * (-0.011111111111111112 * (a * (angle_m * pi)));
	else
		tmp = b * (0.011111111111111112 * (b * (angle_m * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 4e-217], N[(a * N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(0.011111111111111112 * N[(b * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.00000000000000033e-217

   1. Initial program 59.6%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 58.3%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 63.6%

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

   if 4.00000000000000033e-217 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 39.8%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 9.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 9.5%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 57.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 4 \cdot 10^{-217}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-257}:\\ \;\;\;\;a \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-257)
    (* a (* -0.011111111111111112 (* a (* angle_m PI))))
    (* 0.011111111111111112 (* angle_m (* PI (* b b)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-257) {
		tmp = a * (-0.011111111111111112 * (a * (angle_m * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * (b * b)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-257) {
		tmp = a * (-0.011111111111111112 * (a * (angle_m * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * (b * b)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-257:
		tmp = a * (-0.011111111111111112 * (a * (angle_m * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * (b * b)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-257)
		tmp = Float64(a * Float64(-0.011111111111111112 * Float64(a * Float64(angle_m * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b * b))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-257)
		tmp = a * (-0.011111111111111112 * (a * (angle_m * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * (b * b)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-257], N[(a * N[(-0.011111111111111112 * N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999998e-258

   1. Initial program 53.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 52.1%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 60.8%

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

   if -9.9999999999999998e-258 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 48.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 54.8%

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

Alternative 8: 57.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-257}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle\_m \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b 2.0) (pow a 2.0)) -1e-257)
    (* -0.011111111111111112 (* a (* PI (* angle_m a))))
    (* 0.011111111111111112 (* angle_m (* PI (* b b)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -1e-257) {
		tmp = -0.011111111111111112 * (a * (((double) M_PI) * (angle_m * a)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * (b * b)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -1e-257) {
		tmp = -0.011111111111111112 * (a * (Math.PI * (angle_m * a)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * (b * b)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -1e-257:
		tmp = -0.011111111111111112 * (a * (math.pi * (angle_m * a)))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * (b * b)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -1e-257)
		tmp = Float64(-0.011111111111111112 * Float64(a * Float64(pi * Float64(angle_m * a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b * b))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -1e-257)
		tmp = -0.011111111111111112 * (a * (pi * (angle_m * a)));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * (b * b)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -1e-257], N[(-0.011111111111111112 * N[(a * N[(Pi * N[(angle$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -9.9999999999999998e-258

   1. Initial program 53.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 52.1%

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

   13. Applied rewrites 60.8%

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

   if -9.9999999999999998e-258 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 48.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 54.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -1 \cdot 10^{-257}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* (+ b a) (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e-20)
      (* (- b a) (* (+ b a) (* angle_m (* PI 0.011111111111111112))))
      (if (<= (/ angle_m 180.0) 5e+203)
        (* (sin (* 0.011111111111111112 (* angle_m PI))) t_0)
        (* angle_m (* PI (* 0.011111111111111112 t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e-20) {
		tmp = (b - a) * ((b + a) * (angle_m * (((double) M_PI) * 0.011111111111111112)));
	} else if ((angle_m / 180.0) <= 5e+203) {
		tmp = sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * t_0;
	} else {
		tmp = angle_m * (((double) M_PI) * (0.011111111111111112 * t_0));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = (b + a) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 4e-20) {
		tmp = (b - a) * ((b + a) * (angle_m * (Math.PI * 0.011111111111111112)));
	} else if ((angle_m / 180.0) <= 5e+203) {
		tmp = Math.sin((0.011111111111111112 * (angle_m * Math.PI))) * t_0;
	} else {
		tmp = angle_m * (Math.PI * (0.011111111111111112 * t_0));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = (b + a) * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 4e-20:
		tmp = (b - a) * ((b + a) * (angle_m * (math.pi * 0.011111111111111112)))
	elif (angle_m / 180.0) <= 5e+203:
		tmp = math.sin((0.011111111111111112 * (angle_m * math.pi))) * t_0
	else:
		tmp = angle_m * (math.pi * (0.011111111111111112 * t_0))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(Float64(b + a) * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-20)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(angle_m * Float64(pi * 0.011111111111111112))));
	elseif (Float64(angle_m / 180.0) <= 5e+203)
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * t_0);
	else
		tmp = Float64(angle_m * Float64(pi * Float64(0.011111111111111112 * t_0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = (b + a) * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-20)
		tmp = (b - a) * ((b + a) * (angle_m * (pi * 0.011111111111111112)));
	elseif ((angle_m / 180.0) <= 5e+203)
		tmp = sin((0.011111111111111112 * (angle_m * pi))) * t_0;
	else
		tmp = angle_m * (pi * (0.011111111111111112 * t_0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-20], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+203], N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(angle$95$m * N[(Pi * N[(0.011111111111111112 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999978e-20

   1. Initial program 60.0%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 74.5%

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

   if 3.99999999999999978e-20 < (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999994e203

   1. Initial program 27.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

   4. Applied rewrites 43.0%

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

   if 4.99999999999999994e203 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 20.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{-7}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(-a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e-7)
    (*
     (+ b a)
     (*
      (- b a)
      (*
       angle_m
       (fma
        (* -2.2862368541380886e-7 (* angle_m angle_m))
        (* PI (* PI PI))
        (* PI 0.011111111111111112)))))
    (if (<= (/ angle_m 180.0) 2e+176)
      (* (sin (* 0.011111111111111112 (* angle_m PI))) (- (* a a)))
      (* (* (+ b a) (- b a)) (* angle_m (* PI 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e-7) {
		tmp = (b + a) * ((b - a) * (angle_m * fma((-2.2862368541380886e-7 * (angle_m * angle_m)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (((double) M_PI) * 0.011111111111111112))));
	} else if ((angle_m / 180.0) <= 2e+176) {
		tmp = sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * -(a * a);
	} else {
		tmp = ((b + a) * (b - a)) * (angle_m * (((double) M_PI) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e-7)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(angle_m * fma(Float64(-2.2862368541380886e-7 * Float64(angle_m * angle_m)), Float64(pi * Float64(pi * pi)), Float64(pi * 0.011111111111111112)))));
	elseif (Float64(angle_m / 180.0) <= 2e+176)
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * Float64(-Float64(a * a)));
	else
		tmp = Float64(Float64(Float64(b + a) * Float64(b - a)) * Float64(angle_m * Float64(pi * 0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e-7], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(-2.2862368541380886e-7 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+176], N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(a * a), $MachinePrecision])), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.9999999999999995e-8

   1. Initial program 60.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.0

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 70.5

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

   9. Applied rewrites 70.5%

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

   if 9.9999999999999995e-8 < (/.f64 angle #s(literal 180 binary64)) < 2e176

   1. Initial program 24.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.4

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

   7. Applied rewrites 32.4%

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

   if 2e176 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 22.5%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 34.4

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

   5. Applied rewrites 34.4%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-7}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e+203)
    (* (+ b a) (* (- b a) (sin (* 0.011111111111111112 (* angle_m PI)))))
    (* angle_m (* PI (* 0.011111111111111112 (* (+ b a) (- b a))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+203) {
		tmp = (b + a) * ((b - a) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else {
		tmp = angle_m * (((double) M_PI) * (0.011111111111111112 * ((b + a) * (b - a))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+203) {
		tmp = (b + a) * ((b - a) * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	} else {
		tmp = angle_m * (Math.PI * (0.011111111111111112 * ((b + a) * (b - a))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e+203:
		tmp = (b + a) * ((b - a) * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	else:
		tmp = angle_m * (math.pi * (0.011111111111111112 * ((b + a) * (b - a))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+203)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	else
		tmp = Float64(angle_m * Float64(pi * Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(b - a)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+203)
		tmp = (b + a) * ((b - a) * sin((0.011111111111111112 * (angle_m * pi))));
	else
		tmp = angle_m * (pi * (0.011111111111111112 * ((b + a) * (b - a))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+203], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle$95$m * N[(Pi * N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999994e203

   1. Initial program 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.2%

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

   if 4.99999999999999994e203 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 20.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e+203)
    (* (+ b a) (* (- b a) (sin (* angle_m (* PI 0.011111111111111112)))))
    (* angle_m (* PI (* 0.011111111111111112 (* (+ b a) (- b a))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+203) {
		tmp = (b + a) * ((b - a) * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = angle_m * (((double) M_PI) * (0.011111111111111112 * ((b + a) * (b - a))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+203) {
		tmp = (b + a) * ((b - a) * Math.sin((angle_m * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = angle_m * (Math.PI * (0.011111111111111112 * ((b + a) * (b - a))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e+203:
		tmp = (b + a) * ((b - a) * math.sin((angle_m * (math.pi * 0.011111111111111112))))
	else:
		tmp = angle_m * (math.pi * (0.011111111111111112 * ((b + a) * (b - a))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+203)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(angle_m * Float64(pi * Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(b - a)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+203)
		tmp = (b + a) * ((b - a) * sin((angle_m * (pi * 0.011111111111111112))));
	else
		tmp = angle_m * (pi * (0.011111111111111112 * ((b + a) * (b - a))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+203], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle$95$m * N[(Pi * N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999994e203

   1. Initial program 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 70.6

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

   6. Applied rewrites 70.6%

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

   if 4.99999999999999994e203 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 20.7%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 40000000000:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle\_m \cdot angle\_m\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 40000000000.0)
    (*
     (+ b a)
     (*
      (- b a)
      (*
       angle_m
       (fma
        (* -2.2862368541380886e-7 (* angle_m angle_m))
        (* PI (* PI PI))
        (* PI 0.011111111111111112)))))
    (* (* (+ b a) (- b a)) (* angle_m (* PI 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 40000000000.0) {
		tmp = (b + a) * ((b - a) * (angle_m * fma((-2.2862368541380886e-7 * (angle_m * angle_m)), (((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = ((b + a) * (b - a)) * (angle_m * (((double) M_PI) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 40000000000.0)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(angle_m * fma(Float64(-2.2862368541380886e-7 * Float64(angle_m * angle_m)), Float64(pi * Float64(pi * pi)), Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(Float64(b + a) * Float64(b - a)) * Float64(angle_m * Float64(pi * 0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 40000000000.0], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(-2.2862368541380886e-7 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4e10

   1. Initial program 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.8

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

   6. Applied rewrites 78.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 70.1

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

   9. Applied rewrites 70.1%

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

   if 4e10 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 23.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 25.4

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

   5. Applied rewrites 25.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 40000000000:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e-20)
    (* (- b a) (* (+ b a) (* angle_m (* PI 0.011111111111111112))))
    (* angle_m (* PI (* 0.011111111111111112 (* (+ b a) (- b a))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-20) {
		tmp = (b - a) * ((b + a) * (angle_m * (((double) M_PI) * 0.011111111111111112)));
	} else {
		tmp = angle_m * (((double) M_PI) * (0.011111111111111112 * ((b + a) * (b - a))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e-20) {
		tmp = (b - a) * ((b + a) * (angle_m * (Math.PI * 0.011111111111111112)));
	} else {
		tmp = angle_m * (Math.PI * (0.011111111111111112 * ((b + a) * (b - a))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e-20:
		tmp = (b - a) * ((b + a) * (angle_m * (math.pi * 0.011111111111111112)))
	else:
		tmp = angle_m * (math.pi * (0.011111111111111112 * ((b + a) * (b - a))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e-20)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(angle_m * Float64(pi * 0.011111111111111112))));
	else
		tmp = Float64(angle_m * Float64(pi * Float64(0.011111111111111112 * Float64(Float64(b + a) * Float64(b - a)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e-20)
		tmp = (b - a) * ((b + a) * (angle_m * (pi * 0.011111111111111112)));
	else
		tmp = angle_m * (pi * (0.011111111111111112 * ((b + a) * (b - a))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e-20], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle$95$m * N[(Pi * N[(0.011111111111111112 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 3.99999999999999978e-20

   1. Initial program 60.0%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 74.5%

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

   if 3.99999999999999978e-20 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 25.4%

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 28.1

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

   5. Applied rewrites 28.1%

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

   7. Applied rewrites 28.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.1% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

3. 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

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. difference-of-squares N/A

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

   14. lower-*.f64 N/A

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

   15. lower-+.f64 N/A

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

   16. lower--.f64 53.6

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

5. Applied rewrites 53.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.2%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 36.2%

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

13. Applied rewrites 39.5%

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

14. Final simplification 39.5%

    \[\leadsto -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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)))))