ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.8%

Time: 34.5s

Alternatives: 13

Speedup: 32.2×

• 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: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+49}:\\ \;\;\;\;\left(\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+173}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\pi \cdot angle\_m}{180}\right)\right) \cdot t\_0\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \sqrt{{\left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (*
       (* (- b_m a_m) (sin (* PI (* angle_m 0.011111111111111112))))
       (+ b_m a_m))
      (if (<= (/ angle_m 180.0) 4e+173)
        (* (* 2.0 (sin (/ (* PI angle_m) 180.0))) t_0)
        (if (<= (/ angle_m 180.0) 2e+229)
          (* (sin (* 0.011111111111111112 (* PI angle_m))) t_0)
          (*
           (+ b_m a_m)
           (sqrt
            (pow
             (* (- b_m a_m) (sin (* angle_m (* PI 0.011111111111111112))))
             2.0)))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = ((b_m - a_m) * sin((((double) M_PI) * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	} else if ((angle_m / 180.0) <= 4e+173) {
		tmp = (2.0 * sin(((((double) M_PI) * angle_m) / 180.0))) * t_0;
	} else if ((angle_m / 180.0) <= 2e+229) {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle_m))) * t_0;
	} else {
		tmp = (b_m + a_m) * sqrt(pow(((b_m - a_m) * sin((angle_m * (((double) M_PI) * 0.011111111111111112)))), 2.0));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = ((b_m - a_m) * Math.sin((Math.PI * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	} else if ((angle_m / 180.0) <= 4e+173) {
		tmp = (2.0 * Math.sin(((Math.PI * angle_m) / 180.0))) * t_0;
	} else if ((angle_m / 180.0) <= 2e+229) {
		tmp = Math.sin((0.011111111111111112 * (Math.PI * angle_m))) * t_0;
	} else {
		tmp = (b_m + a_m) * Math.sqrt(Math.pow(((b_m - a_m) * Math.sin((angle_m * (Math.PI * 0.011111111111111112)))), 2.0));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 1e+49:
		tmp = ((b_m - a_m) * math.sin((math.pi * (angle_m * 0.011111111111111112)))) * (b_m + a_m)
	elif (angle_m / 180.0) <= 4e+173:
		tmp = (2.0 * math.sin(((math.pi * angle_m) / 180.0))) * t_0
	elif (angle_m / 180.0) <= 2e+229:
		tmp = math.sin((0.011111111111111112 * (math.pi * angle_m))) * t_0
	else:
		tmp = (b_m + a_m) * math.sqrt(math.pow(((b_m - a_m) * math.sin((angle_m * (math.pi * 0.011111111111111112)))), 2.0))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))) * Float64(b_m + a_m));
	elseif (Float64(angle_m / 180.0) <= 4e+173)
		tmp = Float64(Float64(2.0 * sin(Float64(Float64(pi * angle_m) / 180.0))) * t_0);
	elseif (Float64(angle_m / 180.0) <= 2e+229)
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle_m))) * t_0);
	else
		tmp = Float64(Float64(b_m + a_m) * sqrt((Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))) ^ 2.0)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+49)
		tmp = ((b_m - a_m) * sin((pi * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	elseif ((angle_m / 180.0) <= 4e+173)
		tmp = (2.0 * sin(((pi * angle_m) / 180.0))) * t_0;
	elseif ((angle_m / 180.0) <= 2e+229)
		tmp = sin((0.011111111111111112 * (pi * angle_m))) * t_0;
	else
		tmp = (b_m + a_m) * sqrt((((b_m - a_m) * sin((angle_m * (pi * 0.011111111111111112)))) ^ 2.0));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+173], N[(N[(2.0 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+229], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999946e48

   1. Initial program 57.4%

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

      1. associate-*l* 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-*l* 57.4%

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

   3. Simplified 57.4%

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

      1. unpow2 57.4%

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

      2. unpow2 57.4%

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

      3. difference-of-squares 64.6%

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

   6. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 71.5%

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

      2. flip3-+ 24.6%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 21.3%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 21.3%

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

      2. associate-/l* 24.7%

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

      3. *-commutative 24.7%

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

      4. associate-*l* 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in b around 0 72.2%

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

       1. *-un-lft-identity 72.2%

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

       2. associate-*r* 71.7%

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

       3. *-commutative 71.7%

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

       4. associate-*l* 72.8%

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

   13. Applied egg-rr 72.8%

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

   if 9.99999999999999946e48 < (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000001e173

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. *-commutative 28.9%

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

      3. associate-*l* 28.9%

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

   3. Simplified 28.9%

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

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. difference-of-squares 32.6%

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

   6. Applied egg-rr 32.6%

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

   7. Taylor expanded in angle around 0 25.5%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 27.7%

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

   9. Applied egg-rr 27.7%

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

   if 4.0000000000000001e173 < (/.f64 angle #s(literal 180 binary64)) < 2e229

   1. Initial program 14.5%

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

      1. associate-*l* 14.5%

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

      2. *-commutative 14.5%

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

      3. associate-*l* 14.5%

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

   3. Simplified 14.5%

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

      1. unpow2 14.5%

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

      2. unpow2 14.5%

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

      3. difference-of-squares 14.5%

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

   6. Applied egg-rr 14.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 14.5%

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

      2. flip3-+ 5.0%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 5.1%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 3.5%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 3.5%

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

      2. associate-/l* 4.6%

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

      3. *-commutative 4.6%

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

      4. associate-*l* 4.5%

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

   10. Simplified 4.5%

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

   11. Taylor expanded in b around 0 12.9%

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

   12. Taylor expanded in angle around inf 21.7%

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

       1. +-commutative 21.7%

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

       2. *-commutative 21.7%

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

       3. +-commutative 21.7%

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

   14. Simplified 21.7%

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

   if 2e229 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 17.1%

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

      1. associate-*l* 17.1%

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

      2. *-commutative 17.1%

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

      3. associate-*l* 17.1%

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

   3. Simplified 17.1%

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

      1. unpow2 17.1%

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

      2. unpow2 17.1%

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

      3. difference-of-squares 17.1%

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

   6. Applied egg-rr 17.1%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 17.1%

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

      2. flip3-+ 4.6%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 3.7%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 2.6%

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

      2. associate-/l* 3.5%

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

      3. *-commutative 3.5%

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

      4. associate-*l* 4.6%

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

   10. Simplified 4.6%

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

   11. Taylor expanded in b around 0 17.1%

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

       1. add-sqr-sqrt 6.8%

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

       2. sqrt-unprod 39.4%

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

       3. pow2 39.4%

          \[\leadsto \sqrt{\color{blue}{{\left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}^{2}}} \cdot \left(a + b\right) \]

   13. Applied egg-rr 39.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+173}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \sqrt{{\left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a\_m}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq -1 \cdot 10^{+274}:\\ \;\;\;\;\left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(\sqrt[3]{{\pi}^{3}} \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b\_m \cdot \left(1 + \frac{a\_m}{b\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (*
    angle_s
    (if (<=
         (* (* (* 2.0 (- (pow b_m 2.0) (pow a_m 2.0))) (sin t_0)) (cos t_0))
         -1e+274)
      (*
       (*
        (- b_m a_m)
        (sin (* angle_m (* (cbrt (pow PI 3.0)) 0.011111111111111112))))
       (* b_m (+ 1.0 (/ a_m b_m))))
      (*
       (* (- b_m a_m) (sin (* PI (* angle_m 0.011111111111111112))))
       (+ b_m a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a_m, 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+274) {
		tmp = ((b_m - a_m) * sin((angle_m * (cbrt(pow(((double) M_PI), 3.0)) * 0.011111111111111112)))) * (b_m * (1.0 + (a_m / b_m)));
	} else {
		tmp = ((b_m - a_m) * sin((((double) M_PI) * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = Math.PI * (angle_m / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a_m, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= -1e+274) {
		tmp = ((b_m - a_m) * Math.sin((angle_m * (Math.cbrt(Math.pow(Math.PI, 3.0)) * 0.011111111111111112)))) * (b_m * (1.0 + (a_m / b_m)));
	} else {
		tmp = ((b_m - a_m) * Math.sin((Math.PI * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	}
	return angle_s * tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a_m ^ 2.0))) * sin(t_0)) * cos(t_0)) <= -1e+274)
		tmp = Float64(Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(cbrt((pi ^ 3.0)) * 0.011111111111111112)))) * Float64(b_m * Float64(1.0 + Float64(a_m / b_m))));
	else
		tmp = Float64(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))) * Float64(b_m + a_m));
	end
	return Float64(angle_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], -1e+274], N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m * N[(1.0 + N[(a$95$m / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < -9.99999999999999921e273

   1. Initial program 39.5%

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

      1. associate-*l* 39.5%

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

      2. *-commutative 39.5%

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

      3. associate-*l* 39.5%

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

   3. Simplified 39.5%

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

      1. unpow2 39.5%

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

      2. unpow2 39.5%

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

      3. difference-of-squares 39.5%

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

   6. Applied egg-rr 39.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 46.3%

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

      2. flip3-+ 0.7%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 0.7%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 0.7%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 0.7%

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

      2. associate-/l* 0.7%

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

      3. *-commutative 0.7%

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

      4. associate-*l* 0.7%

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

   10. Simplified 0.7%

       \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \frac{{b}^{3} + {a}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 0.8%

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

       2. pow3 0.8%

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

   12. Applied egg-rr 0.8%

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

   13. Taylor expanded in b around inf 61.8%

       \[\leadsto \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\sqrt[3]{{\pi}^{3}} \cdot 0.011111111111111112\right)\right)\right) \cdot \color{blue}{\left(b \cdot \left(1 + \frac{a}{b}\right)\right)} \]

   if -9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 52.2%

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

      1. associate-*l* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-*l* 52.2%

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

   3. Simplified 52.2%

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

      1. unpow2 52.2%

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

      2. unpow2 52.2%

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

      3. difference-of-squares 59.7%

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

   6. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 64.7%

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

      2. flip3-+ 25.2%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 21.8%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 21.7%

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

      2. associate-/l* 25.2%

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

      3. *-commutative 25.2%

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

      4. associate-*l* 25.4%

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

   10. Simplified 25.4%

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

   11. Taylor expanded in b around 0 63.0%

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

       1. *-un-lft-identity 63.0%

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

       2. associate-*r* 63.9%

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

       3. *-commutative 63.9%

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

       4. associate-*l* 64.3%

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

   13. Applied egg-rr 64.3%

       \[\leadsto \left(\left(b - a\right) \cdot \color{blue}{\left(1 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)}\right) \cdot \left(a + b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -1 \cdot 10^{+274}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\sqrt[3]{{\pi}^{3}} \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b \cdot \left(1 + \frac{a}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+49}:\\ \;\;\;\;\left(\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+173}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle\_m \cdot \frac{\pi}{180}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (*
       (* (- b_m a_m) (sin (* PI (* angle_m 0.011111111111111112))))
       (+ b_m a_m))
      (if (<= (/ angle_m 180.0) 4e+173)
        (* (* 2.0 (sin (* angle_m (/ PI 180.0)))) t_0)
        (* (sin (* 0.011111111111111112 (* PI angle_m))) t_0))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = ((b_m - a_m) * sin((((double) M_PI) * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	} else if ((angle_m / 180.0) <= 4e+173) {
		tmp = (2.0 * sin((angle_m * (((double) M_PI) / 180.0)))) * t_0;
	} else {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle_m))) * t_0;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = ((b_m - a_m) * Math.sin((Math.PI * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	} else if ((angle_m / 180.0) <= 4e+173) {
		tmp = (2.0 * Math.sin((angle_m * (Math.PI / 180.0)))) * t_0;
	} else {
		tmp = Math.sin((0.011111111111111112 * (Math.PI * angle_m))) * t_0;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 1e+49:
		tmp = ((b_m - a_m) * math.sin((math.pi * (angle_m * 0.011111111111111112)))) * (b_m + a_m)
	elif (angle_m / 180.0) <= 4e+173:
		tmp = (2.0 * math.sin((angle_m * (math.pi / 180.0)))) * t_0
	else:
		tmp = math.sin((0.011111111111111112 * (math.pi * angle_m))) * t_0
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))) * Float64(b_m + a_m));
	elseif (Float64(angle_m / 180.0) <= 4e+173)
		tmp = Float64(Float64(2.0 * sin(Float64(angle_m * Float64(pi / 180.0)))) * t_0);
	else
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle_m))) * t_0);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+49)
		tmp = ((b_m - a_m) * sin((pi * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	elseif ((angle_m / 180.0) <= 4e+173)
		tmp = (2.0 * sin((angle_m * (pi / 180.0)))) * t_0;
	else
		tmp = sin((0.011111111111111112 * (pi * angle_m))) * t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+173], N[(N[(2.0 * N[Sin[N[(angle$95$m * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999946e48

   1. Initial program 57.4%

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

      1. associate-*l* 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-*l* 57.4%

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

   3. Simplified 57.4%

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

      1. unpow2 57.4%

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

      2. unpow2 57.4%

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

      3. difference-of-squares 64.6%

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

   6. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 71.5%

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

      2. flip3-+ 24.6%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 21.3%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 21.3%

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

      2. associate-/l* 24.7%

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

      3. *-commutative 24.7%

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

      4. associate-*l* 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in b around 0 72.2%

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

       1. *-un-lft-identity 72.2%

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

       2. associate-*r* 71.7%

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

       3. *-commutative 71.7%

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

       4. associate-*l* 72.8%

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

   13. Applied egg-rr 72.8%

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

   if 9.99999999999999946e48 < (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000001e173

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. *-commutative 28.9%

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

      3. associate-*l* 28.9%

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

   3. Simplified 28.9%

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

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. difference-of-squares 32.6%

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

   6. Applied egg-rr 32.6%

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

   7. Taylor expanded in angle around 0 25.5%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 27.7%

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

   9. Applied egg-rr 27.7%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)} \cdot 1\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative 27.7%

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

       2. associate-/l* 30.8%

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

   11. Simplified 30.8%

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

   if 4.0000000000000001e173 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.0%

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

      1. associate-*l* 16.0%

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

      2. *-commutative 16.0%

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

      3. associate-*l* 16.0%

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

   3. Simplified 16.0%

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

      1. unpow2 16.0%

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

      2. unpow2 16.0%

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

      3. difference-of-squares 16.0%

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

   6. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 16.0%

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

      2. flip3-+ 4.8%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 4.3%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 3.0%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 3.0%

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

      2. associate-/l* 3.9%

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

      3. *-commutative 3.9%

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

      4. associate-*l* 4.6%

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

   10. Simplified 4.6%

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

   11. Taylor expanded in b around 0 15.3%

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

   12. Taylor expanded in angle around inf 25.5%

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

       1. +-commutative 25.5%

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

       2. *-commutative 25.5%

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

       3. +-commutative 25.5%

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

   14. Simplified 25.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+173}:\\ \;\;\;\;\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 10^{+49}:\\ \;\;\;\;\left(\left(b\_m - a\_m\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b\_m + a\_m\right)\\ \mathbf{elif}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+173}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\pi \cdot angle\_m}{180}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (let* ((t_0 (* (- b_m a_m) (+ b_m a_m))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 1e+49)
      (*
       (* (- b_m a_m) (sin (* PI (* angle_m 0.011111111111111112))))
       (+ b_m a_m))
      (if (<= (/ angle_m 180.0) 4e+173)
        (* (* 2.0 (sin (/ (* PI angle_m) 180.0))) t_0)
        (* (sin (* 0.011111111111111112 (* PI angle_m))) t_0))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = ((b_m - a_m) * sin((((double) M_PI) * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	} else if ((angle_m / 180.0) <= 4e+173) {
		tmp = (2.0 * sin(((((double) M_PI) * angle_m) / 180.0))) * t_0;
	} else {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle_m))) * t_0;
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double t_0 = (b_m - a_m) * (b_m + a_m);
	double tmp;
	if ((angle_m / 180.0) <= 1e+49) {
		tmp = ((b_m - a_m) * Math.sin((Math.PI * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	} else if ((angle_m / 180.0) <= 4e+173) {
		tmp = (2.0 * Math.sin(((Math.PI * angle_m) / 180.0))) * t_0;
	} else {
		tmp = Math.sin((0.011111111111111112 * (Math.PI * angle_m))) * t_0;
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	t_0 = (b_m - a_m) * (b_m + a_m)
	tmp = 0
	if (angle_m / 180.0) <= 1e+49:
		tmp = ((b_m - a_m) * math.sin((math.pi * (angle_m * 0.011111111111111112)))) * (b_m + a_m)
	elif (angle_m / 180.0) <= 4e+173:
		tmp = (2.0 * math.sin(((math.pi * angle_m) / 180.0))) * t_0
	else:
		tmp = math.sin((0.011111111111111112 * (math.pi * angle_m))) * t_0
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	t_0 = Float64(Float64(b_m - a_m) * Float64(b_m + a_m))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 1e+49)
		tmp = Float64(Float64(Float64(b_m - a_m) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))) * Float64(b_m + a_m));
	elseif (Float64(angle_m / 180.0) <= 4e+173)
		tmp = Float64(Float64(2.0 * sin(Float64(Float64(pi * angle_m) / 180.0))) * t_0);
	else
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle_m))) * t_0);
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	t_0 = (b_m - a_m) * (b_m + a_m);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 1e+49)
		tmp = ((b_m - a_m) * sin((pi * (angle_m * 0.011111111111111112)))) * (b_m + a_m);
	elseif ((angle_m / 180.0) <= 4e+173)
		tmp = (2.0 * sin(((pi * angle_m) / 180.0))) * t_0;
	else
		tmp = sin((0.011111111111111112 * (pi * angle_m))) * t_0;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := Block[{t$95$0 = N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 1e+49], N[(N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+173], N[(N[(2.0 * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999946e48

   1. Initial program 57.4%

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

      1. associate-*l* 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-*l* 57.4%

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

   3. Simplified 57.4%

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

      1. unpow2 57.4%

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

      2. unpow2 57.4%

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

      3. difference-of-squares 64.6%

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

   6. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 71.5%

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

      2. flip3-+ 24.6%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 21.3%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 21.3%

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

      2. associate-/l* 24.7%

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

      3. *-commutative 24.7%

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

      4. associate-*l* 24.8%

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

   10. Simplified 24.8%

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

   11. Taylor expanded in b around 0 72.2%

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

       1. *-un-lft-identity 72.2%

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

       2. associate-*r* 71.7%

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

       3. *-commutative 71.7%

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

       4. associate-*l* 72.8%

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

   13. Applied egg-rr 72.8%

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

   if 9.99999999999999946e48 < (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000001e173

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. *-commutative 28.9%

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

      3. associate-*l* 28.9%

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

   3. Simplified 28.9%

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

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. difference-of-squares 32.6%

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

   6. Applied egg-rr 32.6%

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

   7. Taylor expanded in angle around 0 25.5%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 27.7%

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

   9. Applied egg-rr 27.7%

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

   if 4.0000000000000001e173 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 16.0%

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

      1. associate-*l* 16.0%

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

      2. *-commutative 16.0%

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

      3. associate-*l* 16.0%

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

   3. Simplified 16.0%

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

      1. unpow2 16.0%

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

      2. unpow2 16.0%

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

      3. difference-of-squares 16.0%

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

   6. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 16.0%

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

      2. flip3-+ 4.8%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 4.3%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 3.0%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 3.0%

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

      2. associate-/l* 3.9%

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

      3. *-commutative 3.9%

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

      4. associate-*l* 4.6%

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

   10. Simplified 4.6%

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

   11. Taylor expanded in b around 0 15.3%

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

   12. Taylor expanded in angle around inf 25.5%

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

       1. +-commutative 25.5%

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

       2. *-commutative 25.5%

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

       3. +-commutative 25.5%

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

   14. Simplified 25.5%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{+49}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot \left(b + a\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{+173}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2.9 \cdot 10^{+92}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;angle\_m \leq 1.85 \cdot 10^{+223}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(b\_m \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2.9e+92)
    (* (+ b_m a_m) (* 0.011111111111111112 (* (- b_m a_m) (* PI angle_m))))
    (if (<= angle_m 1.85e+223)
      (* (+ b_m a_m) (* b_m (sin (* 0.011111111111111112 (* PI angle_m)))))
      (* 0.011111111111111112 (* angle_m (* (pow b_m 2.0) PI)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2.9e+92) {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (((double) M_PI) * angle_m)));
	} else if (angle_m <= 1.85e+223) {
		tmp = (b_m + a_m) * (b_m * sin((0.011111111111111112 * (((double) M_PI) * angle_m))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (pow(b_m, 2.0) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2.9e+92) {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (Math.PI * angle_m)));
	} else if (angle_m <= 1.85e+223) {
		tmp = (b_m + a_m) * (b_m * Math.sin((0.011111111111111112 * (Math.PI * angle_m))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.pow(b_m, 2.0) * Math.PI));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 2.9e+92:
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (math.pi * angle_m)))
	elif angle_m <= 1.85e+223:
		tmp = (b_m + a_m) * (b_m * math.sin((0.011111111111111112 * (math.pi * angle_m))))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pow(b_m, 2.0) * math.pi))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 2.9e+92)
		tmp = Float64(Float64(b_m + a_m) * Float64(0.011111111111111112 * Float64(Float64(b_m - a_m) * Float64(pi * angle_m))));
	elseif (angle_m <= 1.85e+223)
		tmp = Float64(Float64(b_m + a_m) * Float64(b_m * sin(Float64(0.011111111111111112 * Float64(pi * angle_m)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64((b_m ^ 2.0) * pi)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 2.9e+92)
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (pi * angle_m)));
	elseif (angle_m <= 1.85e+223)
		tmp = (b_m + a_m) * (b_m * sin((0.011111111111111112 * (pi * angle_m))));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b_m ^ 2.0) * pi));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2.9e+92], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 1.85e+223], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(b$95$m * N[Sin[N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if angle < 2.9000000000000001e92

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-*l* 56.6%

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

   3. Simplified 56.6%

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

      1. unpow2 56.6%

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

      2. unpow2 56.6%

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

      3. difference-of-squares 63.5%

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

   6. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 70.2%

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

      2. flip3-+ 24.0%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 20.7%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 20.7%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 20.7%

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

      2. associate-/l* 23.9%

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

      3. *-commutative 23.9%

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

      4. associate-*l* 24.1%

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

   10. Simplified 24.1%

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

   11. Taylor expanded in b around 0 70.9%

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

   12. Taylor expanded in angle around 0 64.0%

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

       1. associate-*r* 64.0%

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

   14. Simplified 64.0%

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

   if 2.9000000000000001e92 < angle < 1.8500000000000001e223

   1. Initial program 23.9%

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

      1. associate-*l* 23.9%

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

      2. *-commutative 23.9%

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

      3. associate-*l* 23.9%

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

   3. Simplified 23.9%

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

      1. unpow2 23.9%

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

      2. unpow2 23.9%

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

      3. difference-of-squares 27.2%

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

   6. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 27.2%

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

      2. flip3-+ 3.9%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 3.9%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 4.8%

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

      2. associate-/l* 5.3%

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

      3. *-commutative 5.3%

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

      4. associate-*l* 4.5%

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

   10. Simplified 4.5%

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

   11. Taylor expanded in b around 0 23.6%

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

   12. Taylor expanded in b around inf 24.8%

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

       1. *-commutative 24.8%

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

   14. Simplified 24.8%

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

   if 1.8500000000000001e223 < angle

   1. Initial program 15.8%

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

      1. associate-*l* 15.8%

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

      2. *-commutative 15.8%

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

      3. associate-*l* 15.8%

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

   3. Simplified 15.8%

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

      1. unpow2 15.8%

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

      2. unpow2 15.8%

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

      3. difference-of-squares 15.8%

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

   6. Applied egg-rr 15.8%

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

   7. Taylor expanded in angle around 0 26.9%

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

   8. Taylor expanded in angle around 0 26.2%

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

   9. Taylor expanded in a around 0 37.2%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.9 \cdot 10^{+92}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.85 \cdot 10^{+223}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 4.8 \cdot 10^{-34}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 4.8e-34)
    (* (+ b_m a_m) (* 0.011111111111111112 (* (- b_m a_m) (* PI angle_m))))
    (*
     (sin (* 0.011111111111111112 (* PI angle_m)))
     (* (- b_m a_m) (+ b_m a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 4.8e-34) {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (((double) M_PI) * angle_m)));
	} else {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle_m))) * ((b_m - a_m) * (b_m + a_m));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 4.8e-34) {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (Math.PI * angle_m)));
	} else {
		tmp = Math.sin((0.011111111111111112 * (Math.PI * angle_m))) * ((b_m - a_m) * (b_m + a_m));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 4.8e-34:
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (math.pi * angle_m)))
	else:
		tmp = math.sin((0.011111111111111112 * (math.pi * angle_m))) * ((b_m - a_m) * (b_m + a_m))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 4.8e-34)
		tmp = Float64(Float64(b_m + a_m) * Float64(0.011111111111111112 * Float64(Float64(b_m - a_m) * Float64(pi * angle_m))));
	else
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle_m))) * Float64(Float64(b_m - a_m) * Float64(b_m + a_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 4.8e-34)
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (pi * angle_m)));
	else
		tmp = sin((0.011111111111111112 * (pi * angle_m))) * ((b_m - a_m) * (b_m + a_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 4.8e-34], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.79999999999999982e-34

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. *-commutative 57.3%

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

      3. associate-*l* 57.3%

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

   3. Simplified 57.3%

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

      1. unpow2 57.3%

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

      2. unpow2 57.3%

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

      3. difference-of-squares 62.6%

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

   6. Applied egg-rr 62.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 70.7%

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

      2. flip3-+ 27.1%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 23.4%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 23.4%

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

      2. associate-/l* 27.1%

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

      3. *-commutative 27.1%

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

      4. associate-*l* 27.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in b around 0 71.8%

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

   12. Taylor expanded in angle around 0 66.5%

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

       1. associate-*r* 66.5%

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

   14. Simplified 66.5%

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

   if 4.79999999999999982e-34 < angle

   1. Initial program 33.8%

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

      1. associate-*l* 33.8%

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

      2. *-commutative 33.8%

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

      3. associate-*l* 33.8%

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

   3. Simplified 33.8%

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

      1. unpow2 33.8%

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

      2. unpow2 33.8%

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

      3. difference-of-squares 40.9%

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

   6. Applied egg-rr 40.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 40.9%

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

      2. flip3-+ 5.9%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 5.3%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 5.1%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 5.1%

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

      2. associate-/l* 5.9%

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

      3. *-commutative 5.9%

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

      4. associate-*l* 6.5%

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

   10. Simplified 6.5%

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

   11. Taylor expanded in b around 0 39.0%

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

   12. Taylor expanded in angle around inf 41.0%

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

       1. +-commutative 41.0%

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

       2. *-commutative 41.0%

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

       3. +-commutative 41.0%

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

   14. Simplified 41.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 4.8 \cdot 10^{-34}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right) \cdot \left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 2e+164)
    (*
     (+ b_m a_m)
     (* (- b_m a_m) (sin (* angle_m (* PI 0.011111111111111112)))))
    (*
     (sin (* 0.011111111111111112 (* PI angle_m)))
     (* (- b_m a_m) (+ b_m a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2e+164) {
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle_m))) * ((b_m - a_m) * (b_m + a_m));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 2e+164) {
		tmp = (b_m + a_m) * ((b_m - a_m) * Math.sin((angle_m * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = Math.sin((0.011111111111111112 * (Math.PI * angle_m))) * ((b_m - a_m) * (b_m + a_m));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 2e+164:
		tmp = (b_m + a_m) * ((b_m - a_m) * math.sin((angle_m * (math.pi * 0.011111111111111112))))
	else:
		tmp = math.sin((0.011111111111111112 * (math.pi * angle_m))) * ((b_m - a_m) * (b_m + a_m))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 2e+164)
		tmp = Float64(Float64(b_m + a_m) * Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle_m))) * Float64(Float64(b_m - a_m) * Float64(b_m + a_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 2e+164)
		tmp = (b_m + a_m) * ((b_m - a_m) * sin((angle_m * (pi * 0.011111111111111112))));
	else
		tmp = sin((0.011111111111111112 * (pi * angle_m))) * ((b_m - a_m) * (b_m + a_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 2e+164], 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], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2e164

   1. Initial program 54.4%

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

      1. associate-*l* 54.4%

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

      2. *-commutative 54.4%

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

      3. associate-*l* 54.4%

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

   3. Simplified 54.4%

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

      1. unpow2 54.4%

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

      2. unpow2 54.4%

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

      3. difference-of-squares 61.3%

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

   6. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 67.5%

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

      2. flip3-+ 22.5%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 19.5%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 19.5%

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

      2. associate-/l* 22.5%

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

      3. *-commutative 22.5%

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

      4. associate-*l* 22.7%

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

   10. Simplified 22.7%

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

   11. Taylor expanded in b around 0 67.8%

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

   if 2e164 < angle

   1. Initial program 17.6%

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

      1. associate-*l* 17.6%

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

      2. *-commutative 17.6%

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

      3. associate-*l* 17.6%

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

   3. Simplified 17.6%

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

      1. unpow2 17.6%

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

      2. unpow2 17.6%

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

      3. difference-of-squares 17.6%

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

   6. Applied egg-rr 17.6%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 17.6%

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

      2. flip3-+ 4.5%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 4.1%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 3.3%

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

      2. associate-/l* 4.2%

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

      3. *-commutative 4.2%

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

      4. associate-*l* 4.7%

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

   10. Simplified 4.7%

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

   11. Taylor expanded in b around 0 17.4%

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

   12. Taylor expanded in angle around inf 26.4%

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

       1. +-commutative 26.4%

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

       2. *-commutative 26.4%

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

       3. +-commutative 26.4%

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

   14. Simplified 26.4%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 3 \cdot 10^{-86}:\\ \;\;\;\;a\_m \cdot \left(\left(b\_m - a\_m\right) \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 3e-86)
    (* a_m (* (- b_m a_m) (sin (* angle_m (* PI 0.011111111111111112)))))
    (* (+ b_m a_m) (* 0.011111111111111112 (* (- b_m a_m) (* PI angle_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 3e-86) {
		tmp = a_m * ((b_m - a_m) * sin((angle_m * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (((double) M_PI) * angle_m)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 3e-86) {
		tmp = a_m * ((b_m - a_m) * Math.sin((angle_m * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (Math.PI * angle_m)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if b_m <= 3e-86:
		tmp = a_m * ((b_m - a_m) * math.sin((angle_m * (math.pi * 0.011111111111111112))))
	else:
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (math.pi * angle_m)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 3e-86)
		tmp = Float64(a_m * Float64(Float64(b_m - a_m) * sin(Float64(angle_m * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(0.011111111111111112 * Float64(Float64(b_m - a_m) * Float64(pi * angle_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (b_m <= 3e-86)
		tmp = a_m * ((b_m - a_m) * sin((angle_m * (pi * 0.011111111111111112))));
	else
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (pi * angle_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 3e-86], N[(a$95$m * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[Sin[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000001e-86

   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. Step-by-step derivation

      1. associate-*l* 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*l* 50.6%

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

   3. Simplified 50.6%

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

      1. unpow2 50.6%

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

      2. unpow2 50.6%

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

      3. difference-of-squares 56.1%

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

   6. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 61.6%

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

      2. flip3-+ 21.1%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 18.1%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 18.1%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 18.1%

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

      2. associate-/l* 21.2%

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

      3. *-commutative 21.2%

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

      4. associate-*l* 21.4%

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

   10. Simplified 21.4%

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

   11. Taylor expanded in b around 0 44.5%

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

   if 3.0000000000000001e-86 < b

   1. Initial program 46.9%

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

      1. associate-*l* 46.9%

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

      2. *-commutative 46.9%

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

      3. associate-*l* 46.9%

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

   3. Simplified 46.9%

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

      1. unpow2 46.9%

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

      2. unpow2 46.9%

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

      3. difference-of-squares 53.9%

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

   6. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 59.1%

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

      2. flip3-+ 17.7%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 15.7%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 15.5%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 15.5%

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

      2. associate-/l* 17.4%

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

      3. *-commutative 17.4%

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

      4. associate-*l* 17.6%

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

   10. Simplified 17.6%

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

   11. Taylor expanded in b around 0 60.5%

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

   12. Taylor expanded in angle around 0 54.7%

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

       1. associate-*r* 54.7%

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

   14. Simplified 54.7%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-86}:\\ \;\;\;\;a \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left({b\_m}^{2} \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 1.4e+145)
    (* (+ b_m a_m) (* 0.011111111111111112 (* (- b_m a_m) (* PI angle_m))))
    (* 0.011111111111111112 (* angle_m (* (pow b_m 2.0) PI))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 1.4e+145) {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (((double) M_PI) * angle_m)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (pow(b_m, 2.0) * ((double) M_PI)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 1.4e+145) {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (Math.PI * angle_m)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.pow(b_m, 2.0) * Math.PI));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 1.4e+145:
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (math.pi * angle_m)))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pow(b_m, 2.0) * math.pi))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 1.4e+145)
		tmp = Float64(Float64(b_m + a_m) * Float64(0.011111111111111112 * Float64(Float64(b_m - a_m) * Float64(pi * angle_m))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64((b_m ^ 2.0) * pi)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 1.4e+145)
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (pi * angle_m)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b_m ^ 2.0) * pi));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 1.4e+145], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.3999999999999999e145

   1. Initial program 54.8%

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

      1. associate-*l* 54.8%

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

      2. *-commutative 54.8%

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

      3. associate-*l* 54.8%

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

   3. Simplified 54.8%

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

      1. unpow2 54.8%

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

      2. unpow2 54.8%

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

      3. difference-of-squares 61.4%

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

   6. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 67.8%

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

      2. flip3-+ 23.0%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 19.9%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 19.9%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 19.9%

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

      2. associate-/l* 23.0%

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

      3. *-commutative 23.0%

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

      4. associate-*l* 23.2%

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

   10. Simplified 23.2%

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

   11. Taylor expanded in b around 0 68.5%

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

   12. Taylor expanded in angle around 0 61.7%

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

       1. associate-*r* 61.7%

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

   14. Simplified 61.7%

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

   if 1.3999999999999999e145 < angle

   1. Initial program 21.1%

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

      1. associate-*l* 21.1%

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

      2. *-commutative 21.1%

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

      3. associate-*l* 21.1%

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

   3. Simplified 21.1%

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

      1. unpow2 21.1%

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

      2. unpow2 21.1%

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

      3. difference-of-squares 23.6%

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

   6. Applied egg-rr 23.6%

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

   7. Taylor expanded in angle around 0 26.5%

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

   8. Taylor expanded in angle around 0 19.4%

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

   9. Taylor expanded in a around 0 30.0%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. associate-*l* 49.5%

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

   2. *-commutative 49.5%

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

   3. associate-*l* 49.5%

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

3. Simplified 49.5%

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

   1. unpow2 49.5%

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

   2. unpow2 49.5%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
7. Step-by-step derivation

   1. associate-*l* 60.9%

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

   2. flip3-+ 20.1%

      \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

   3. associate-*l/ 17.5%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

8. Applied egg-rr 17.4%

   \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
9. Step-by-step derivation

   1. *-commutative 17.4%

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

   2. associate-/l* 20.1%

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

   3. *-commutative 20.1%

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

   4. associate-*l* 20.3%

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

10. Simplified 20.3%

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

11. Taylor expanded in b around 0 61.1%

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

    1. *-un-lft-identity 61.1%

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

    2. associate-*r* 62.2%

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

    3. *-commutative 62.2%

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

    4. associate-*l* 62.2%

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

13. Applied egg-rr 62.2%

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

14. Final simplification 62.2%

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

Alternative 11: 64.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b\_m - a\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= angle_m 5e-49)
    (* (+ b_m a_m) (* 0.011111111111111112 (* angle_m (* PI (- b_m a_m)))))
    (* 0.011111111111111112 (* angle_m (* PI (* (- b_m a_m) (+ b_m a_m))))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 5e-49) {
		tmp = (b_m + a_m) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (b_m - a_m))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((b_m - a_m) * (b_m + a_m))));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (angle_m <= 5e-49) {
		tmp = (b_m + a_m) * (0.011111111111111112 * (angle_m * (Math.PI * (b_m - a_m))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((b_m - a_m) * (b_m + a_m))));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if angle_m <= 5e-49:
		tmp = (b_m + a_m) * (0.011111111111111112 * (angle_m * (math.pi * (b_m - a_m))))
	else:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((b_m - a_m) * (b_m + a_m))))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (angle_m <= 5e-49)
		tmp = Float64(Float64(b_m + a_m) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b_m - a_m)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b_m - a_m) * Float64(b_m + a_m)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (angle_m <= 5e-49)
		tmp = (b_m + a_m) * (0.011111111111111112 * (angle_m * (pi * (b_m - a_m))));
	else
		tmp = 0.011111111111111112 * (angle_m * (pi * ((b_m - a_m) * (b_m + a_m))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 5e-49], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b$95$m - a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.9999999999999999e-49

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. difference-of-squares 62.0%

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

   6. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 70.2%

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

      2. flip3-+ 26.9%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 23.2%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 23.2%

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

      2. associate-/l* 26.9%

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

      3. *-commutative 26.9%

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

      4. associate-*l* 26.9%

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

   10. Simplified 26.9%

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

   11. Taylor expanded in b around 0 71.3%

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

   12. Taylor expanded in angle around 0 65.9%

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

   if 4.9999999999999999e-49 < angle

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. *-commutative 36.0%

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

      3. associate-*l* 36.0%

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

   3. Simplified 36.0%

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

      1. unpow2 36.0%

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

      2. unpow2 36.0%

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

      3. difference-of-squares 42.9%

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

   6. Applied egg-rr 42.9%

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

   7. Taylor expanded in angle around 0 37.0%

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

   8. Taylor expanded in angle around 0 34.2%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b\_m \leq 2 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(b\_m - a\_m\right) \cdot \left(b\_m + a\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m + a\_m\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b\_m - a\_m\right) \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a_m b_m angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b_m 2e-151)
    (* 0.011111111111111112 (* angle_m (* PI (* (- b_m a_m) (+ b_m a_m)))))
    (* (+ b_m a_m) (* 0.011111111111111112 (* (- b_m a_m) (* PI angle_m)))))))

C

a_m = fabs(a);
b_m = fabs(b);
angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 2e-151) {
		tmp = 0.011111111111111112 * (angle_m * (((double) M_PI) * ((b_m - a_m) * (b_m + a_m))));
	} else {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (((double) M_PI) * angle_m)));
	}
	return angle_s * tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a_m, double b_m, double angle_m) {
	double tmp;
	if (b_m <= 2e-151) {
		tmp = 0.011111111111111112 * (angle_m * (Math.PI * ((b_m - a_m) * (b_m + a_m))));
	} else {
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (Math.PI * angle_m)));
	}
	return angle_s * tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a_m, b_m, angle_m):
	tmp = 0
	if b_m <= 2e-151:
		tmp = 0.011111111111111112 * (angle_m * (math.pi * ((b_m - a_m) * (b_m + a_m))))
	else:
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (math.pi * angle_m)))
	return angle_s * tmp

Julia

a_m = abs(a)
b_m = abs(b)
angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0
	if (b_m <= 2e-151)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(b_m - a_m) * Float64(b_m + a_m)))));
	else
		tmp = Float64(Float64(b_m + a_m) * Float64(0.011111111111111112 * Float64(Float64(b_m - a_m) * Float64(pi * angle_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

a_m = abs(a);
b_m = abs(b);
angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a_m, b_m, angle_m)
	tmp = 0.0;
	if (b_m <= 2e-151)
		tmp = 0.011111111111111112 * (angle_m * (pi * ((b_m - a_m) * (b_m + a_m))));
	else
		tmp = (b_m + a_m) * (0.011111111111111112 * ((b_m - a_m) * (pi * angle_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a$95$m_, b$95$m_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b$95$m, 2e-151], N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(b$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m + a$95$m), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(b$95$m - a$95$m), $MachinePrecision] * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999999e-151

   1. Initial program 51.6%

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

      1. associate-*l* 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*l* 51.6%

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

   3. Simplified 51.6%

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

      1. unpow2 51.6%

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

      2. unpow2 51.6%

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

      3. difference-of-squares 57.7%

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

   6. Applied egg-rr 57.7%

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

   7. Taylor expanded in angle around 0 54.8%

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

   8. Taylor expanded in angle around 0 53.4%

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

   if 1.9999999999999999e-151 < b

   1. Initial program 45.7%

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

      1. associate-*l* 45.7%

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

      2. *-commutative 45.7%

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

      3. associate-*l* 45.7%

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

   3. Simplified 45.7%

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

      1. unpow2 45.7%

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

      2. unpow2 45.7%

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

      3. difference-of-squares 51.4%

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

   6. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*l* 55.5%

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

      2. flip3-+ 20.8%

         \[\leadsto \color{blue}{\frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \]

      3. associate-*l/ 19.2%

         \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}{b \cdot b + \left(a \cdot a - b \cdot a\right)}} \]

   8. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{\frac{\left({b}^{3} + {a}^{3}\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(a - b\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 18.9%

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

      2. associate-/l* 20.5%

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

      3. *-commutative 20.5%

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

      4. associate-*l* 21.1%

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

   10. Simplified 21.1%

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

   11. Taylor expanded in b around 0 57.2%

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

   12. Taylor expanded in angle around 0 50.1%

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

       1. associate-*r* 50.2%

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

   14. Simplified 50.2%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-151}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.7% accurate, 32.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. associate-*l* 49.5%

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

   2. *-commutative 49.5%

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

   3. associate-*l* 49.5%

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

3. Simplified 49.5%

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

   1. unpow2 49.5%

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

   2. unpow2 49.5%

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

   3. difference-of-squares 55.5%

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

6. Applied egg-rr 55.5%

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

7. Taylor expanded in angle around 0 54.2%

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

8. Taylor expanded in angle around 0 50.8%

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

9. Final simplification 50.8%

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

Reproduce

herbie shell --seed 2024085 
(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)))))