ab-angle->ABCF B

Percentage Accurate: 53.4% → 66.3%

Time: 16.7s

Alternatives: 16

Speedup: 6.0×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{-1}{\left(b\_m + a\_m\right) \cdot \left(\sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(a\_m - b\_m\right)\right)}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\sqrt{\pi} \cdot \left(angle\_m \cdot \sqrt{\pi}\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0
         (/
          -1.0
          (*
           (+ b_m a_m)
           (* (sin (* angle_m (* PI 0.011111111111111112))) (- a_m b_m))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+161)
      (*
       (+ b_m a_m)
       (*
        (- b_m a_m)
        (sin (* (* (sqrt PI) (* angle_m (sqrt PI))) 0.011111111111111112))))
      (if (<= (/ angle_m 180.0) 5e+255)
        (pow (* t_0 t_0) -0.5)
        (* -0.011111111111111112 (* a_m (* angle_m (* (+ b_m a_m) PI)))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = -1.0 / ((b_m + a_m) * (sin((angle_m * (((double) M_PI) * 0.011111111111111112))) * (a_m - b_m)));
	double tmp;
	if ((angle_m / 180.0) <= 4e+161) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin(((sqrt(((double) M_PI)) * (angle_m * sqrt(((double) M_PI)))) * 0.011111111111111112)));
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = pow((t_0 * t_0), -0.5);
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = -1.0 / ((b_m + a_m) * (Math.sin((angle_m * (Math.PI * 0.011111111111111112))) * (a_m - b_m)));
	double tmp;
	if ((angle_m / 180.0) <= 4e+161) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin(((Math.sqrt(Math.PI) * (angle_m * Math.sqrt(Math.PI))) * 0.011111111111111112)));
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = Math.pow((t_0 * t_0), -0.5);
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = -1.0 / ((b_m + a_m) * (math.sin((angle_m * (math.pi * 0.011111111111111112))) * (a_m - b_m)))
	tmp = 0
	if (angle_m / 180.0) <= 4e+161:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin(((math.sqrt(math.pi) * (angle_m * math.sqrt(math.pi))) * 0.011111111111111112)))
	elif (angle_m / 180.0) <= 5e+255:
		tmp = math.pow((t_0 * t_0), -0.5)
	else:
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * math.pi)))
	return angle_s * tmp

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(-1.0 / Float64(Float64(b_m + a_m) * Float64(sin(Float64(angle_m * Float64(pi * 0.011111111111111112))) * Float64(a_m - b_m))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+161)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(sqrt(pi) * Float64(angle_m * sqrt(pi))) * 0.011111111111111112))));
	elseif (Float64(angle_m / 180.0) <= 5e+255)
		tmp = Float64(t_0 * t_0) ^ -0.5;
	else
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(Float64(b_m + a_m) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = -1.0 / ((b_m + a_m) * (sin((angle_m * (pi * 0.011111111111111112))) * (a_m - b_m)));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+161)
		tmp = (b_m + a_m) * ((b_m - a_m) * sin(((sqrt(pi) * (angle_m * sqrt(pi))) * 0.011111111111111112)));
	elseif ((angle_m / 180.0) <= 5e+255)
		tmp = (t_0 * t_0) ^ -0.5;
	else
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(-1.0 / N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a$95$m - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+161], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(angle$95$m * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+255], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], -0.5], $MachinePrecision], N[(-0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e161

   1. Initial program 53.8%

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

   4. Applied rewrites 71.6%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 71.5

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

   6. Applied rewrites 71.5%

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

   if 4.0000000000000002e161 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e255

   1. Initial program 28.2%

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

   4. Applied rewrites 28.9%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 36.7

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

   6. Applied rewrites 36.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

   8. Applied rewrites 47.4%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} \cdot \frac{1}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}\right)}^{-0.5}} \]

   if 5.0000000000000002e255 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.3%

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 16.6

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-+.f64 24.0

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

   10. Applied rewrites 24.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+161}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\sqrt{\pi} \cdot \left(angle \cdot \sqrt{\pi}\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;{\left(\frac{-1}{\left(b + a\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(a - b\right)\right)} \cdot \frac{-1}{\left(b + a\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(a - b\right)\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\\ t_1 := angle\_m \cdot \sqrt{\pi}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\left(\sqrt{\pi} \cdot t\_1\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(0.011111111111111112 \cdot \left(t\_1 \cdot \sqrt{t\_0 \cdot t\_0}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (pow (cbrt (sqrt PI)) 3.0)) (t_1 (* angle_m (sqrt PI))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+80)
      (*
       (+ b_m a_m)
       (* (- b_m a_m) (sin (* (* (sqrt PI) t_1) 0.011111111111111112))))
      (*
       (+ b_m a_m)
       (*
        (- b_m a_m)
        (sin (* 0.011111111111111112 (* t_1 (sqrt (* t_0 t_0)))))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = pow(cbrt(sqrt(((double) M_PI))), 3.0);
	double t_1 = angle_m * sqrt(((double) M_PI));
	double tmp;
	if ((angle_m / 180.0) <= 5e+80) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin(((sqrt(((double) M_PI)) * t_1) * 0.011111111111111112)));
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((0.011111111111111112 * (t_1 * sqrt((t_0 * t_0))))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0);
	double t_1 = angle_m * Math.sqrt(Math.PI);
	double tmp;
	if ((angle_m / 180.0) <= 5e+80) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin(((Math.sqrt(Math.PI) * t_1) * 0.011111111111111112)));
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((0.011111111111111112 * (t_1 * Math.sqrt((t_0 * t_0))))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = cbrt(sqrt(pi)) ^ 3.0
	t_1 = Float64(angle_m * sqrt(pi))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+80)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(Float64(sqrt(pi) * t_1) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(0.011111111111111112 * Float64(t_1 * sqrt(Float64(t_0 * t_0)))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(angle$95$m * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+80], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[(N[Sqrt[Pi], $MachinePrecision] * t$95$1), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(t$95$1 * N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999961e80

   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

   4. Applied rewrites 73.7%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 74.2

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

   6. Applied rewrites 74.2%

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

   if 4.99999999999999961e80 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.7%

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

   4. Applied rewrites 36.5%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 36.7

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

   6. Applied rewrites 36.7%

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

      1. add-cube-cbrt N/A

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

      2. pow3 N/A

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

      3. lift-PI.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. cbrt-prod N/A

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

      8. unpow-prod-down N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-cbrt.f64 44.3

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

   8. Applied rewrites 44.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\sqrt{\pi} \cdot \left(angle \cdot \sqrt{\pi}\right)\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\left(angle \cdot \sqrt{\pi}\right) \cdot \sqrt{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b_m 2.0) 2e+52)
    (*
     (+ b_m a_m)
     (* (- b_m a_m) (sin (* PI (* angle_m 0.011111111111111112)))))
    (*
     (+ b_m a_m)
     (*
      (- b_m a_m)
      (*
       angle_m
       (fma
        -2.2862368541380886e-7
        (* (* angle_m angle_m) (* PI (* PI PI)))
        (* PI 0.011111111111111112))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (pow(b_m, 2.0) <= 2e+52) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * (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

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 2e+52)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 2e+52], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 2e52

   1. Initial program 59.1%

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

   4. Applied rewrites 65.9%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 64.2

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

   6. Applied rewrites 64.2%

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

   if 2e52 < (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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 69.4

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

   7. Applied rewrites 69.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 10^{-120}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(-a\_m\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b_m 2.0) 1e-120)
    (* (+ b_m a_m) (* (- a_m) (sin (* 0.011111111111111112 (* angle_m PI)))))
    (*
     (+ b_m a_m)
     (*
      (- b_m a_m)
      (*
       angle_m
       (fma
        -2.2862368541380886e-7
        (* (* angle_m angle_m) (* PI (* PI PI)))
        (* PI 0.011111111111111112))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (pow(b_m, 2.0) <= 1e-120) {
		tmp = (b_m + a_m) * (-a_m * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * (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

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 1e-120)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(-a_m) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 1e-120], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[((-a$95$m) * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 9.99999999999999979e-121

   1. Initial program 59.1%

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 64.9

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

   7. Applied rewrites 64.9%

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

   if 9.99999999999999979e-121 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 45.6%

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

   4. Applied rewrites 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 angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 67.5

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

   7. Applied rewrites 67.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{-120}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(-a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} - {a\_m}^{2} \leq -5 \cdot 10^{-163}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a\_m \cdot \left(angle\_m \cdot a\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (- (pow b_m 2.0) (pow a_m 2.0)) -5e-163)
    (* -0.011111111111111112 (* PI (* a_m (* angle_m a_m))))
    (* (* angle_m 0.011111111111111112) (* PI (* b_m b_m))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((pow(b_m, 2.0) - pow(a_m, 2.0)) <= -5e-163) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (a_m * (angle_m * a_m)));
	} else {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0)) <= -5e-163) {
		tmp = -0.011111111111111112 * (Math.PI * (a_m * (angle_m * a_m)));
	} else {
		tmp = (angle_m * 0.011111111111111112) * (Math.PI * (b_m * b_m));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if (math.pow(b_m, 2.0) - math.pow(a_m, 2.0)) <= -5e-163:
		tmp = -0.011111111111111112 * (math.pi * (a_m * (angle_m * a_m)))
	else:
		tmp = (angle_m * 0.011111111111111112) * (math.pi * (b_m * b_m))
	return angle_s * tmp

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (Float64((b_m ^ 2.0) - (a_m ^ 2.0)) <= -5e-163)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(a_m * Float64(angle_m * a_m))));
	else
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (((b_m ^ 2.0) - (a_m ^ 2.0)) <= -5e-163)
		tmp = -0.011111111111111112 * (pi * (a_m * (angle_m * a_m)));
	else
		tmp = (angle_m * 0.011111111111111112) * (pi * (b_m * b_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision], -5e-163], N[(-0.011111111111111112 * N[(Pi * N[(a$95$m * N[(angle$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -4.99999999999999977e-163

   1. Initial program 51.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 48.5

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

   8. Applied rewrites 48.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.4

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

   10. Applied rewrites 61.4%

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

   if -4.99999999999999977e-163 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 50.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 56.4

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

   8. Applied rewrites 56.4%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{-163}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b\_m}^{2} \leq 10^{-186}:\\ \;\;\;\;a\_m \cdot \left(\left(-a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (pow b_m 2.0) 1e-186)
    (* a_m (* (- a_m) (sin (* PI (* angle_m 0.011111111111111112)))))
    (*
     (+ b_m a_m)
     (*
      (- b_m a_m)
      (*
       angle_m
       (fma
        -2.2862368541380886e-7
        (* (* angle_m angle_m) (* PI (* PI PI)))
        (* PI 0.011111111111111112))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (pow(b_m, 2.0) <= 1e-186) {
		tmp = a_m * (-a_m * sin((((double) M_PI) * (angle_m * 0.011111111111111112))));
	} else {
		tmp = (b_m + a_m) * ((b_m - a_m) * (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

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if ((b_m ^ 2.0) <= 1e-186)
		tmp = Float64(a_m * Float64(Float64(-a_m) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[Power[b$95$m, 2.0], $MachinePrecision], 1e-186], N[(a$95$m * N[((-a$95$m) * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 9.9999999999999991e-187

   1. Initial program 58.4%

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

   4. Applied rewrites 67.9%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 65.9

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

   6. Applied rewrites 65.9%

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

   7. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   8. 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. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.4

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

   9. Applied rewrites 57.4%

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

       1. lift-PI.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sin.f64 N/A

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

       6. distribute-lft-neg-in N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-neg.f64 65.4

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

   11. Applied rewrites 65.4%

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

   if 9.9999999999999991e-187 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 47.0%

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

   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 angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 66.3

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

   7. Applied rewrites 66.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{-186}:\\ \;\;\;\;a \cdot \left(\left(-a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 3 \cdot 10^{+209}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 1.7e+92)
    (*
     (+ b_m a_m)
     (* (- b_m a_m) (sin (* 0.011111111111111112 (* angle_m PI)))))
    (if (<= b_m 3e+209)
      (*
       (+ b_m a_m)
       (*
        (- b_m a_m)
        (*
         angle_m
         (fma
          -2.2862368541380886e-7
          (* (* angle_m angle_m) (* PI (* PI PI)))
          (* PI 0.011111111111111112)))))
      (*
       (+ b_m a_m)
       (*
        (- b_m a_m)
        (sin
         (* (sqrt PI) (* (sqrt PI) (* angle_m 0.011111111111111112))))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 1.7e+92) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else if (b_m <= 3e+209) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (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_m + a_m) * ((b_m - a_m) * sin((sqrt(((double) M_PI)) * (sqrt(((double) M_PI)) * (angle_m * 0.011111111111111112)))));
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 1.7e+92)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	elseif (b_m <= 3e+209)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(sqrt(pi) * Float64(sqrt(pi) * Float64(angle_m * 0.011111111111111112))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 1.7e+92], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3e+209], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < 1.6999999999999999e92

   1. Initial program 55.8%

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

   4. Applied rewrites 66.6%

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

   if 1.6999999999999999e92 < b < 2.99999999999999985e209

   1. Initial program 24.6%

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 74.1

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

   7. Applied rewrites 74.1%

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

   if 2.99999999999999985e209 < b

   1. Initial program 30.6%

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

   4. Applied rewrites 66.6%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l* N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-*.f64 74.9

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

   6. Applied rewrites 74.9%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+209}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a\_m \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \left(\left(b\_m + a\_m\right) \cdot \left(a\_m - b\_m\right)\right)}{b\_m - a\_m}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 5e+97)
    (*
     (+ b_m a_m)
     (* (- b_m a_m) (sin (* 0.011111111111111112 (* angle_m PI)))))
    (/
     (*
      (* a_m (sin (* angle_m (* PI 0.011111111111111112))))
      (* (+ b_m a_m) (- a_m b_m)))
     (- b_m a_m)))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+97) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else {
		tmp = ((a_m * sin((angle_m * (((double) M_PI) * 0.011111111111111112)))) * ((b_m + a_m) * (a_m - b_m))) / (b_m - a_m);
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e+97) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	} else {
		tmp = ((a_m * Math.sin((angle_m * (Math.PI * 0.011111111111111112)))) * ((b_m + a_m) * (a_m - b_m))) / (b_m - a_m);
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e+97:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	else:
		tmp = ((a_m * math.sin((angle_m * (math.pi * 0.011111111111111112)))) * ((b_m + a_m) * (a_m - b_m))) / (b_m - a_m)
	return angle_s * tmp

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+97)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	else
		tmp = Float64(Float64(Float64(a_m * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))) * Float64(Float64(b_m + a_m) * Float64(a_m - b_m))) / Float64(b_m - a_m));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+97)
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((0.011111111111111112 * (angle_m * pi))));
	else
		tmp = ((a_m * sin((angle_m * (pi * 0.011111111111111112)))) * ((b_m + a_m) * (a_m - b_m))) / (b_m - a_m);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+97], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(a$95$m - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999999e97

   1. Initial program 55.6%

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

   4. Applied rewrites 73.6%

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

   if 4.99999999999999999e97 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 29.5%

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

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 27.7

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

   7. Applied rewrites 27.7%

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

      1. flip-+ N/A

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

      2. lift--.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 36.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 66.1%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l* N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-*.f64 66.2

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

6. Applied rewrites 66.2%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 67.6

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

8. Applied rewrites 67.6%

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

9. Final simplification 67.6%

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

Alternative 10: 64.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e+148)
    (*
     (+ b_m a_m)
     (* (- b_m a_m) (sin (* 0.011111111111111112 (* angle_m PI)))))
    (if (<= (/ angle_m 180.0) 5e+255)
      (* (* angle_m 0.011111111111111112) (* PI (* b_m b_m)))
      (* -0.011111111111111112 (* a_m (* angle_m (* (+ b_m a_m) PI))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+148) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((0.011111111111111112 * (angle_m * ((double) M_PI)))));
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m));
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+148) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((0.011111111111111112 * (angle_m * Math.PI))));
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (Math.PI * (b_m * b_m));
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 4e+148:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin((0.011111111111111112 * (angle_m * math.pi))))
	elif (angle_m / 180.0) <= 5e+255:
		tmp = (angle_m * 0.011111111111111112) * (math.pi * (b_m * b_m))
	else:
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * math.pi)))
	return angle_s * tmp

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+148)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(0.011111111111111112 * Float64(angle_m * pi)))));
	elseif (Float64(angle_m / 180.0) <= 5e+255)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m)));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(Float64(b_m + a_m) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+148)
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((0.011111111111111112 * (angle_m * pi))));
	elseif ((angle_m / 180.0) <= 5e+255)
		tmp = (angle_m * 0.011111111111111112) * (pi * (b_m * b_m));
	else
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+148], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+255], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e148

   1. Initial program 54.3%

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

   4. Applied rewrites 71.7%

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

   if 4.0000000000000002e148 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e255

   1. Initial program 25.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3

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

   8. Applied rewrites 46.3%

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

   if 5.0000000000000002e255 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.3%

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 16.6

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-+.f64 24.0

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

   10. Applied rewrites 24.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+148}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 1e+148)
    (*
     (+ b_m a_m)
     (* (- b_m a_m) (sin (* angle_m (* PI 0.011111111111111112)))))
    (if (<= (/ angle_m 180.0) 5e+255)
      (* (* angle_m 0.011111111111111112) (* PI (* b_m b_m)))
      (* -0.011111111111111112 (* a_m (* angle_m (* (+ b_m a_m) PI))))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+148) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m));
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * ((double) M_PI))));
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 1e+148) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((angle_m * (Math.PI * 0.011111111111111112))));
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (Math.PI * (b_m * b_m));
	} else {
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * Math.PI)));
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 1e+148:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin((angle_m * (math.pi * 0.011111111111111112))))
	elif (angle_m / 180.0) <= 5e+255:
		tmp = (angle_m * 0.011111111111111112) * (math.pi * (b_m * b_m))
	else:
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * math.pi)))
	return angle_s * tmp

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+148)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	elseif (Float64(angle_m / 180.0) <= 5e+255)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m)));
	else
		tmp = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(Float64(b_m + a_m) * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+148)
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((angle_m * (pi * 0.011111111111111112))));
	elseif ((angle_m / 180.0) <= 5e+255)
		tmp = (angle_m * 0.011111111111111112) * (pi * (b_m * b_m));
	else
		tmp = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+148], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+255], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e148

   1. Initial program 54.3%

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

   4. Applied rewrites 71.7%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.0

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

   6. Applied rewrites 71.0%

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

   if 1e148 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e255

   1. Initial program 25.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3

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

   8. Applied rewrites 46.3%

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

   if 5.0000000000000002e255 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 31.3%

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 16.6

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-+.f64 24.0

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

   10. Applied rewrites 24.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+148}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+85}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(angle\_m \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* -0.011111111111111112 (* a_m (* angle_m (* (+ b_m a_m) PI))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+85)
      (*
       (+ b_m a_m)
       (*
        (- b_m a_m)
        (*
         angle_m
         (fma
          -2.2862368541380886e-7
          (* (* angle_m angle_m) (* PI (* PI PI)))
          (* PI 0.011111111111111112)))))
      (if (<= (/ angle_m 180.0) 4e+148)
        t_0
        (if (<= (/ angle_m 180.0) 5e+255)
          (* (* angle_m 0.011111111111111112) (* PI (* b_m b_m)))
          t_0))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 1e+85) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (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) <= 4e+148) {
		tmp = t_0;
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m));
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(Float64(b_m + a_m) * pi))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+85)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(angle_m * fma(-2.2862368541380886e-7, Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * pi))), Float64(pi * 0.011111111111111112)))));
	elseif (Float64(angle_m / 180.0) <= 4e+148)
		tmp = t_0;
	elseif (Float64(angle_m / 180.0) <= 5e+255)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m)));
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(-0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+85], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(angle$95$m * N[(-2.2862368541380886e-7 * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+148], t$95$0, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+255], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e85

   1. Initial program 55.1%

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-PI.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if 1e85 < (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e148 or 5.0000000000000002e255 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 37.6%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 30.9

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

   7. Applied rewrites 30.9%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-+.f64 40.4

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

   10. Applied rewrites 40.4%

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

   if 4.0000000000000002e148 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e255

   1. Initial program 25.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3

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

   8. Applied rewrites 46.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+85}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* -0.011111111111111112 (* a_m (* angle_m (* (+ b_m a_m) PI))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+55)
      (* (+ b_m a_m) (* (- b_m a_m) (* 0.011111111111111112 (* angle_m PI))))
      (if (<= (/ angle_m 180.0) 4e+148)
        t_0
        (if (<= (/ angle_m 180.0) 5e+255)
          (* (* angle_m 0.011111111111111112) (* PI (* b_m b_m)))
          t_0))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 4e+55) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else if ((angle_m / 180.0) <= 4e+148) {
		tmp = t_0;
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m));
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * Math.PI)));
	double tmp;
	if ((angle_m / 180.0) <= 4e+55) {
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle_m * Math.PI)));
	} else if ((angle_m / 180.0) <= 4e+148) {
		tmp = t_0;
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (Math.PI * (b_m * b_m));
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * math.pi)))
	tmp = 0
	if (angle_m / 180.0) <= 4e+55:
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle_m * math.pi)))
	elif (angle_m / 180.0) <= 4e+148:
		tmp = t_0
	elif (angle_m / 180.0) <= 5e+255:
		tmp = (angle_m * 0.011111111111111112) * (math.pi * (b_m * b_m))
	else:
		tmp = t_0
	return angle_s * tmp

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(Float64(b_m + a_m) * pi))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+55)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * Float64(0.011111111111111112 * Float64(angle_m * pi))));
	elseif (Float64(angle_m / 180.0) <= 4e+148)
		tmp = t_0;
	elseif (Float64(angle_m / 180.0) <= 5e+255)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m)));
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * pi)));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+55)
		tmp = (b_m + a_m) * ((b_m - a_m) * (0.011111111111111112 * (angle_m * pi)));
	elseif ((angle_m / 180.0) <= 4e+148)
		tmp = t_0;
	elseif ((angle_m / 180.0) <= 5e+255)
		tmp = (angle_m * 0.011111111111111112) * (pi * (b_m * b_m));
	else
		tmp = t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(-0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+55], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+148], t$95$0, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+255], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000004e55

   1. Initial program 54.7%

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

   3. Taylor expanded in angle around 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 58.7

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

   5. Applied rewrites 58.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 71.5

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

   if 4.00000000000000004e55 < (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e148 or 5.0000000000000002e255 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 44.2%

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

   4. Applied rewrites 44.8%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 33.0

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-+.f64 38.4

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

   10. Applied rewrites 38.4%

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

   if 4.0000000000000002e148 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e255

   1. Initial program 25.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3

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

   8. Applied rewrites 46.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a\_m \cdot \left(angle\_m \cdot \left(\left(b\_m + a\_m\right) \cdot \pi\right)\right)\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* -0.011111111111111112 (* a_m (* angle_m (* (+ b_m a_m) PI))))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 4e+55)
      (* (+ b_m a_m) (* (* angle_m 0.011111111111111112) (* (- b_m a_m) PI)))
      (if (<= (/ angle_m 180.0) 4e+148)
        t_0
        (if (<= (/ angle_m 180.0) 5e+255)
          (* (* angle_m 0.011111111111111112) (* PI (* b_m b_m)))
          t_0))))))

C

b_m = fabs(b);
a_m = fabs(a);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 4e+55) {
		tmp = (b_m + a_m) * ((angle_m * 0.011111111111111112) * ((b_m - a_m) * ((double) M_PI)));
	} else if ((angle_m / 180.0) <= 4e+148) {
		tmp = t_0;
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (((double) M_PI) * (b_m * b_m));
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * Math.PI)));
	double tmp;
	if ((angle_m / 180.0) <= 4e+55) {
		tmp = (b_m + a_m) * ((angle_m * 0.011111111111111112) * ((b_m - a_m) * Math.PI));
	} else if ((angle_m / 180.0) <= 4e+148) {
		tmp = t_0;
	} else if ((angle_m / 180.0) <= 5e+255) {
		tmp = (angle_m * 0.011111111111111112) * (Math.PI * (b_m * b_m));
	} else {
		tmp = t_0;
	}
	return angle_s * tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * math.pi)))
	tmp = 0
	if (angle_m / 180.0) <= 4e+55:
		tmp = (b_m + a_m) * ((angle_m * 0.011111111111111112) * ((b_m - a_m) * math.pi))
	elif (angle_m / 180.0) <= 4e+148:
		tmp = t_0
	elif (angle_m / 180.0) <= 5e+255:
		tmp = (angle_m * 0.011111111111111112) * (math.pi * (b_m * b_m))
	else:
		tmp = t_0
	return angle_s * tmp

Julia

b_m = abs(b)
a_m = abs(a)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(-0.011111111111111112 * Float64(a_m * Float64(angle_m * Float64(Float64(b_m + a_m) * pi))))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+55)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(b_m - a_m) * pi)));
	elseif (Float64(angle_m / 180.0) <= 4e+148)
		tmp = t_0;
	elseif (Float64(angle_m / 180.0) <= 5e+255)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(pi * Float64(b_m * b_m)));
	else
		tmp = t_0;
	end
	return Float64(angle_s * tmp)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = -0.011111111111111112 * (a_m * (angle_m * ((b_m + a_m) * pi)));
	tmp = 0.0;
	if ((angle_m / 180.0) <= 4e+55)
		tmp = (b_m + a_m) * ((angle_m * 0.011111111111111112) * ((b_m - a_m) * pi));
	elseif ((angle_m / 180.0) <= 4e+148)
		tmp = t_0;
	elseif ((angle_m / 180.0) <= 5e+255)
		tmp = (angle_m * 0.011111111111111112) * (pi * (b_m * b_m));
	else
		tmp = t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(-0.011111111111111112 * N[(a$95$m * N[(angle$95$m * N[(N[(b$95$m + a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+55], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+148], t$95$0, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+255], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.00000000000000004e55

   1. Initial program 54.7%

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower--.f64 71.5

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

   7. Applied rewrites 71.5%

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

   if 4.00000000000000004e55 < (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000002e148 or 5.0000000000000002e255 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 44.2%

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

   4. Applied rewrites 44.8%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 33.0

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-+.f64 38.4

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

   10. Applied rewrites 38.4%

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

   if 4.0000000000000002e148 < (/.f64 angle #s(literal 180 binary64)) < 5.0000000000000002e255

   1. Initial program 25.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.3

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

   8. Applied rewrites 46.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.5% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(-0.011111111111111112 * N[(Pi * N[(a$95$m * N[(angle$95$m * a$95$m), $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 52.5

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

5. Applied rewrites 52.5%

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

6. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 30.4

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

8. Applied rewrites 30.4%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 34.2

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

10. Applied rewrites 34.2%

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

11. Final simplification 34.2%

    \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right) \]
12. Add Preprocessing

Alternative 16: 38.5% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * N[(-0.011111111111111112 * N[(a$95$m * N[(a$95$m * N[(angle$95$m * Pi), $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 52.5

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

5. Applied rewrites 52.5%

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

6. Taylor expanded in b around 0

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

   1. lower-*.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 30.4

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

8. Applied rewrites 30.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. associate-*l* N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 34.1

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lift-sqrt.f64 N/A

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

   18. lift-sqrt.f64 N/A

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

10. Applied rewrites 34.2%

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

Reproduce

herbie shell --seed 2024212 
(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)))))