ab-angle->ABCF B

Percentage Accurate: 53.5% → 64.4%

Time: 29.3s

Alternatives: 15

Speedup: 5.4×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{\pi}}\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;b \leq 4 \cdot 10^{+248}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(t_1 \cdot {t_1}^{2}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt PI))) (t_1 (* t_0 t_0)))
   (if (<= b 4e+248)
     (*
      (+ b a)
      (*
       (- b a)
       (sin (* 2.0 (* (* t_1 (pow t_1 2.0)) (* 0.005555555555555556 angle))))))
     (*
      (* (* 2.0 (* b b)) (sin (* 0.005555555555555556 (* PI angle))))
      (cos (* PI (/ angle 180.0)))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = cbrt(sqrt(((double) M_PI)));
	double t_1 = t_0 * t_0;
	double tmp;
	if (b <= 4e+248) {
		tmp = (b + a) * ((b - a) * sin((2.0 * ((t_1 * pow(t_1, 2.0)) * (0.005555555555555556 * angle)))));
	} else {
		tmp = ((2.0 * (b * b)) * sin((0.005555555555555556 * (((double) M_PI) * angle)))) * cos((((double) M_PI) * (angle / 180.0)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.sqrt(Math.PI));
	double t_1 = t_0 * t_0;
	double tmp;
	if (b <= 4e+248) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * ((t_1 * Math.pow(t_1, 2.0)) * (0.005555555555555556 * angle)))));
	} else {
		tmp = ((2.0 * (b * b)) * Math.sin((0.005555555555555556 * (Math.PI * angle)))) * Math.cos((Math.PI * (angle / 180.0)));
	}
	return tmp;
}

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = cbrt(sqrt(pi))
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (b <= 4e+248)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(t_1 * (t_1 ^ 2.0)) * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(b * b)) * sin(Float64(0.005555555555555556 * Float64(pi * angle)))) * cos(Float64(pi * Float64(angle / 180.0))));
	end
	return tmp
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[b, 4e+248], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000018e248

   1. Initial program 57.9%

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

      1. *-commutative 57.9%

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

      2. associate-*l* 57.9%

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

      3. associate-*l* 57.9%

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

      4. unpow2 57.9%

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

      5. unpow2 57.9%

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

      6. difference-of-squares 62.2%

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

   3. Simplified 62.2%

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

      1. associate-*l* 69.1%

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

      2. flip-+ 58.0%

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

      3. associate-*l/ 55.9%

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

      4. associate-*l* 55.9%

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

      5. 2-sin 55.9%

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

      6. div-inv 55.2%

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

      7. metadata-eval 55.2%

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

   5. Applied egg-rr 55.2%

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

      1. associate-/l* 57.1%

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

      2. associate-/r/ 57.2%

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

      3. difference-of-squares 61.0%

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

      4. associate-/l* 67.9%

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

      5. *-inverses 67.9%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 67.9%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 67.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 65.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

      2. pow2 65.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   9. Applied egg-rr 65.9%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. pow1/3 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\pi}^{0.3333333333333333}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       2. add-sqr-sqrt 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{0.3333333333333333}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       3. unpow2 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\color{blue}{\left({\left(\sqrt{\pi}\right)}^{2}\right)}}^{0.3333333333333333}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       4. pow-pow 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{\left(2 \cdot 0.3333333333333333\right)}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       5. metadata-eval 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\left(\sqrt{\pi}\right)}^{\color{blue}{0.6666666666666666}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   11. Applied egg-rr 65.9%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{0.6666666666666666}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. metadata-eval 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\left(\sqrt{\pi}\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       2. pow-sqr 68.0%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       3. unpow1/3 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\color{blue}{\sqrt[3]{\sqrt{\pi}}} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       4. unpow1/3 70.3%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   13. Simplified 70.3%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. pow1/3 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\pi}^{0.3333333333333333}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       2. add-sqr-sqrt 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{0.3333333333333333}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       3. unpow2 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\color{blue}{\left({\left(\sqrt{\pi}\right)}^{2}\right)}}^{0.3333333333333333}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       4. pow-pow 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{\left(2 \cdot 0.3333333333333333\right)}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       5. metadata-eval 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\left(\sqrt{\pi}\right)}^{\color{blue}{0.6666666666666666}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   15. Applied egg-rr 70.3%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.6666666666666666}\right)}}^{2} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   16. Step-by-step derivation

       1. metadata-eval 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\left(\sqrt{\pi}\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       2. pow-sqr 68.0%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       3. unpow1/3 65.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\color{blue}{\sqrt[3]{\sqrt{\pi}}} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       4. unpow1/3 70.3%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   17. Simplified 69.7%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}^{2} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   if 4.00000000000000018e248 < b

   1. Initial program 57.1%

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

      1. sub-neg 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. unpow2 57.1%

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

      5. unpow2 57.1%

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

      6. sqr-neg 57.1%

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

      7. unpow2 57.1%

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

      8. unpow2 57.1%

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

      9. sqr-neg 57.1%

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

      10. unpow2 57.1%

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

      11. sub-neg 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in b around inf 92.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow2 92.9%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*r* 92.9%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. *-commutative 92.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. *-commutative 92.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. associate-*r* 85.7%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative 85.7%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Simplified 85.7%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 92.9%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+248}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot {\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}^{2}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]

Alternative 2: 66.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{\pi}}\\ \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\left(t_0 \cdot t_0\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt PI))))
   (if (<= (pow a 2.0) 5e+94)
     (*
      (+ b a)
      (* (- b a) (sin (* 2.0 (* angle (* PI 0.005555555555555556))))))
     (*
      (+ b a)
      (*
       (- b a)
       (sin
        (*
         2.0
         (*
          (* 0.005555555555555556 angle)
          (* (* t_0 t_0) (pow (cbrt PI) 2.0))))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = cbrt(sqrt(((double) M_PI)));
	double tmp;
	if (pow(a, 2.0) <= 5e+94) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (angle * (((double) M_PI) * 0.005555555555555556)))));
	} else {
		tmp = (b + a) * ((b - a) * sin((2.0 * ((0.005555555555555556 * angle) * ((t_0 * t_0) * pow(cbrt(((double) M_PI)), 2.0))))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.sqrt(Math.PI));
	double tmp;
	if (Math.pow(a, 2.0) <= 5e+94) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (angle * (Math.PI * 0.005555555555555556)))));
	} else {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * ((0.005555555555555556 * angle) * ((t_0 * t_0) * Math.pow(Math.cbrt(Math.PI), 2.0))))));
	}
	return tmp;
}

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = cbrt(sqrt(pi))
	tmp = 0.0
	if ((a ^ 2.0) <= 5e+94)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle) * Float64(Float64(t_0 * t_0) * (cbrt(pi) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 5e+94], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 5.0000000000000001e94

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*l* 65.5%

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

      3. associate-*l* 65.5%

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

      4. unpow2 65.5%

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

      5. unpow2 65.5%

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

      6. difference-of-squares 65.5%

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

   3. Simplified 65.5%

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

      1. associate-*l* 69.3%

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

      2. flip-+ 65.6%

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

      3. associate-*l/ 63.1%

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

      4. associate-*l* 63.1%

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

      5. 2-sin 63.1%

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

      6. div-inv 62.7%

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

      7. metadata-eval 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. associate-/l* 64.8%

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

      2. associate-/r/ 64.9%

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

      3. difference-of-squares 64.9%

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

      4. associate-/l* 68.6%

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

      5. *-inverses 68.6%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 68.6%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 68.6%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 68.6%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 68.5%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 68.5%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 69.7%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 69.7%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   if 5.0000000000000001e94 < (pow.f64 a 2)

   1. Initial program 46.7%

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

      1. *-commutative 46.7%

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

      2. associate-*l* 46.7%

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

      3. associate-*l* 46.7%

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

      4. unpow2 46.7%

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

      5. unpow2 46.7%

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

      6. difference-of-squares 59.6%

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

   3. Simplified 59.6%

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

      1. associate-*l* 70.2%

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

      2. flip-+ 46.7%

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

      3. associate-*l/ 45.6%

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

      4. associate-*l* 45.6%

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

      5. 2-sin 45.6%

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

      6. div-inv 44.6%

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

      7. metadata-eval 44.6%

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

   5. Applied egg-rr 44.6%

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

      1. associate-/l* 45.8%

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

      2. associate-/r/ 45.8%

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

      3. difference-of-squares 57.7%

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

      4. associate-/l* 68.3%

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

      5. *-inverses 68.3%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 68.3%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 68.3%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 66.6%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

      2. pow2 66.6%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   9. Applied egg-rr 66.6%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. pow1/3 66.6%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\pi}^{0.3333333333333333}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       2. add-sqr-sqrt 66.6%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{0.3333333333333333}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       3. unpow2 66.6%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\color{blue}{\left({\left(\sqrt{\pi}\right)}^{2}\right)}}^{0.3333333333333333}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       4. pow-pow 66.6%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{\left(2 \cdot 0.3333333333333333\right)}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       5. metadata-eval 66.6%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\left(\sqrt{\pi}\right)}^{\color{blue}{0.6666666666666666}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   11. Applied egg-rr 66.6%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{0.6666666666666666}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. metadata-eval 66.6%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot {\left(\sqrt{\pi}\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       2. pow-sqr 70.1%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       3. unpow1/3 66.6%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\color{blue}{\sqrt[3]{\sqrt{\pi}}} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       4. unpow1/3 75.9%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}\right)\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   13. Simplified 75.9%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{e^{3 \cdot \log \pi}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (pow a 2.0) 2e+120)
   (* (+ b a) (* (- b a) (sin (* 2.0 (* angle (* PI 0.005555555555555556))))))
   (*
    (+ b a)
    (*
     (- b a)
     (sin
      (*
       2.0
       (* (* 0.005555555555555556 angle) (cbrt (exp (* 3.0 (log PI)))))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (pow(a, 2.0) <= 2e+120) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (angle * (((double) M_PI) * 0.005555555555555556)))));
	} else {
		tmp = (b + a) * ((b - a) * sin((2.0 * ((0.005555555555555556 * angle) * cbrt(exp((3.0 * log(((double) M_PI)))))))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(a, 2.0) <= 2e+120) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (angle * (Math.PI * 0.005555555555555556)))));
	} else {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * ((0.005555555555555556 * angle) * Math.cbrt(Math.exp((3.0 * Math.log(Math.PI))))))));
	}
	return tmp;
}

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if ((a ^ 2.0) <= 2e+120)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle) * cbrt(exp(Float64(3.0 * log(pi)))))))));
	end
	return tmp
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[N[Power[a, 2.0], $MachinePrecision], 2e+120], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Exp[N[(3.0 * N[Log[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 2e120

   1. Initial program 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-*l* 65.4%

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

      3. associate-*l* 65.4%

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

      4. unpow2 65.4%

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

      5. unpow2 65.4%

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

      6. difference-of-squares 65.4%

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

   3. Simplified 65.4%

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

      1. associate-*l* 69.0%

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

      2. flip-+ 65.4%

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

      3. associate-*l/ 63.0%

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

      4. associate-*l* 63.0%

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

      5. 2-sin 63.0%

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

      6. div-inv 62.6%

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

      7. metadata-eval 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. associate-/l* 64.7%

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

      2. associate-/r/ 64.8%

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

      3. difference-of-squares 64.8%

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

      4. associate-/l* 68.3%

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

      5. *-inverses 68.3%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 68.3%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 68.3%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 68.3%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 68.3%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 69.4%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 69.4%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   if 2e120 < (pow.f64 a 2)

   1. Initial program 46.0%

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

      1. *-commutative 46.0%

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

      2. associate-*l* 46.0%

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

      3. associate-*l* 46.0%

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

      4. unpow2 46.0%

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

      5. unpow2 46.0%

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

      6. difference-of-squares 59.6%

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

   3. Simplified 59.6%

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

      1. associate-*l* 70.7%

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

      2. flip-+ 46.0%

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

      3. associate-*l/ 44.8%

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

      4. associate-*l* 44.8%

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

      5. 2-sin 44.8%

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

      6. div-inv 43.8%

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

      7. metadata-eval 43.8%

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

   5. Applied egg-rr 43.8%

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

      1. associate-/l* 45.0%

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

      2. associate-/r/ 45.1%

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

      3. difference-of-squares 57.5%

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

      4. associate-/l* 68.7%

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

      5. *-inverses 68.7%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 68.7%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 68.7%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 73.5%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

      2. pow3 73.5%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   9. Applied egg-rr 73.5%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 73.5%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt[3]{\color{blue}{e^{\log \left({\pi}^{3}\right)}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

       2. log-pow 76.4%

          \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt[3]{e^{\color{blue}{3 \cdot \log \pi}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   11. Applied egg-rr 76.4%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt[3]{\color{blue}{e^{3 \cdot \log \pi}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{e^{3 \cdot \log \pi}}\right)\right)\right)\\ \end{array} \]

Alternative 4: 66.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{+217}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{0.005555555555555556 \cdot angle}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.3e+217)
   (* (+ b a) (* (- b a) (sin (* 2.0 (* angle (* PI 0.005555555555555556))))))
   (*
    (+ b a)
    (*
     (- b a)
     (sin (* 2.0 (* PI (pow (cbrt (* 0.005555555555555556 angle)) 3.0))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.3e+217) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (angle * (((double) M_PI) * 0.005555555555555556)))));
	} else {
		tmp = (b + a) * ((b - a) * sin((2.0 * (((double) M_PI) * pow(cbrt((0.005555555555555556 * angle)), 3.0)))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.3e+217) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (angle * (Math.PI * 0.005555555555555556)))));
	} else {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.PI * Math.pow(Math.cbrt((0.005555555555555556 * angle)), 3.0)))));
	}
	return tmp;
}

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.3e+217)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(angle * Float64(pi * 0.005555555555555556))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * (cbrt(Float64(0.005555555555555556 * angle)) ^ 3.0))))));
	end
	return tmp
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 4.3e+217], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[Power[N[Power[N[(0.005555555555555556 * angle), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.3000000000000001e217

   1. Initial program 59.9%

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

      1. *-commutative 59.9%

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

      2. associate-*l* 59.9%

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

      3. associate-*l* 59.9%

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

      4. unpow2 59.9%

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

      5. unpow2 59.9%

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

      6. difference-of-squares 64.2%

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

   3. Simplified 64.2%

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

      1. associate-*l* 70.4%

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

      2. flip-+ 59.9%

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

      3. associate-*l/ 57.9%

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

      4. associate-*l* 57.9%

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

      5. 2-sin 57.9%

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

      6. div-inv 57.6%

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

      7. metadata-eval 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. associate-/l* 59.5%

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

      2. associate-/r/ 59.5%

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

      3. difference-of-squares 63.4%

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

      4. associate-/l* 69.6%

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

      5. *-inverses 69.6%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 69.6%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 69.6%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 69.7%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 69.7%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 69.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 69.9%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   if 4.3000000000000001e217 < a

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-*l* 29.8%

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

      3. associate-*l* 29.8%

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

      4. unpow2 29.8%

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

      5. unpow2 29.8%

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

      6. difference-of-squares 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*l* 58.7%

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

      2. flip-+ 29.8%

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

      3. associate-*l/ 29.8%

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

      4. associate-*l* 29.8%

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

      5. 2-sin 29.8%

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

      6. div-inv 23.9%

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

      7. metadata-eval 23.9%

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

   5. Applied egg-rr 23.9%

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

      1. associate-/l* 23.9%

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

      2. associate-/r/ 23.9%

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

      3. difference-of-squares 42.0%

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

      4. associate-/l* 52.8%

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

      5. *-inverses 52.8%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 52.8%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 52.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 52.8%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 70.4%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{0.005555555555555556 \cdot angle} \cdot \sqrt[3]{0.005555555555555556 \cdot angle}\right) \cdot \sqrt[3]{0.005555555555555556 \cdot angle}\right)}\right)\right)\right) \]

      2. pow3 70.4%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{{\left(\sqrt[3]{0.005555555555555556 \cdot angle}\right)}^{3}}\right)\right)\right) \]

   9. Applied egg-rr 70.4%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{{\left(\sqrt[3]{0.005555555555555556 \cdot angle}\right)}^{3}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{+217}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{0.005555555555555556 \cdot angle}\right)}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 5: 65.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+252}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.6e+252)
   (*
    (+ b a)
    (*
     (- b a)
     (sin (* 2.0 (* (* 0.005555555555555556 angle) (cbrt (pow PI 3.0)))))))
   (*
    (* (* 2.0 (* b b)) (sin (* 0.005555555555555556 (* PI angle))))
    (cos (* PI (/ angle 180.0))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.6e+252) {
		tmp = (b + a) * ((b - a) * sin((2.0 * ((0.005555555555555556 * angle) * cbrt(pow(((double) M_PI), 3.0))))));
	} else {
		tmp = ((2.0 * (b * b)) * sin((0.005555555555555556 * (((double) M_PI) * angle)))) * cos((((double) M_PI) * (angle / 180.0)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.6e+252) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * ((0.005555555555555556 * angle) * Math.cbrt(Math.pow(Math.PI, 3.0))))));
	} else {
		tmp = ((2.0 * (b * b)) * Math.sin((0.005555555555555556 * (Math.PI * angle)))) * Math.cos((Math.PI * (angle / 180.0)));
	}
	return tmp;
}

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.6e+252)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle) * cbrt((pi ^ 3.0)))))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(b * b)) * sin(Float64(0.005555555555555556 * Float64(pi * angle)))) * cos(Float64(pi * Float64(angle / 180.0))));
	end
	return tmp
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 5.6e+252], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.60000000000000007e252

   1. Initial program 57.9%

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

      1. *-commutative 57.9%

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

      2. associate-*l* 57.9%

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

      3. associate-*l* 57.9%

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

      4. unpow2 57.9%

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

      5. unpow2 57.9%

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

      6. difference-of-squares 62.2%

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

   3. Simplified 62.2%

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

      1. associate-*l* 69.1%

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

      2. flip-+ 58.0%

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

      3. associate-*l/ 55.9%

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

      4. associate-*l* 55.9%

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

      5. 2-sin 55.9%

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

      6. div-inv 55.2%

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

      7. metadata-eval 55.2%

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

   5. Applied egg-rr 55.2%

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

      1. associate-/l* 57.1%

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

      2. associate-/r/ 57.2%

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

      3. difference-of-squares 61.0%

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

      4. associate-/l* 67.9%

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

      5. *-inverses 67.9%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 67.9%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 67.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 70.0%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

      2. pow3 70.0%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   9. Applied egg-rr 70.0%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]

   if 5.60000000000000007e252 < b

   1. Initial program 57.1%

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

      1. sub-neg 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. unpow2 57.1%

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

      5. unpow2 57.1%

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

      6. sqr-neg 57.1%

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

      7. unpow2 57.1%

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

      8. unpow2 57.1%

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

      9. sqr-neg 57.1%

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

      10. unpow2 57.1%

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

      11. sub-neg 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in b around inf 92.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left({b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   5. Step-by-step derivation

      1. unpow2 92.9%

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*r* 92.9%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. *-commutative 92.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. *-commutative 92.9%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. associate-*r* 85.7%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative 85.7%

         \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Simplified 85.7%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 92.9%

      \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+252}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\\ \end{array} \]

Alternative 6: 66.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -2e+159)
   (* 0.011111111111111112 (* angle (* PI (* (+ b a) (- b a)))))
   (* (+ b a) (* (- b a) (sin (* (* PI angle) 0.011111111111111112))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+159) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * ((b + a) * (b - a))));
	} else {
		tmp = (b + a) * ((b - a) * sin(((((double) M_PI) * angle) * 0.011111111111111112)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+159) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * ((b + a) * (b - a))));
	} else {
		tmp = (b + a) * ((b - a) * Math.sin(((Math.PI * angle) * 0.011111111111111112)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -2e+159:
		tmp = 0.011111111111111112 * (angle * (math.pi * ((b + a) * (b - a))))
	else:
		tmp = (b + a) * ((b - a) * math.sin(((math.pi * angle) * 0.011111111111111112)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+159)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b + a) * Float64(b - a)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(Float64(pi * angle) * 0.011111111111111112))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+159)
		tmp = 0.011111111111111112 * (angle * (pi * ((b + a) * (b - a))));
	else
		tmp = (b + a) * ((b - a) * sin(((pi * angle) * 0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+159], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < -1.9999999999999999e159

   1. Initial program 21.9%

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

      1. *-commutative 21.9%

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

      2. associate-*l* 21.9%

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

      3. associate-*l* 21.9%

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

      4. unpow2 21.9%

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

      5. unpow2 21.9%

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

      6. difference-of-squares 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in angle around 0 40.7%

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

   if -1.9999999999999999e159 < (/.f64 angle 180)

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*l* 62.3%

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

      3. associate-*l* 62.3%

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

      4. unpow2 62.3%

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

      5. unpow2 62.3%

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

      6. difference-of-squares 67.7%

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

   3. Simplified 67.7%

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

      1. associate-*l* 75.1%

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

      2. flip-+ 62.4%

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

      3. associate-*l/ 60.4%

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

      4. associate-*l* 60.4%

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

      5. 2-sin 60.4%

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

      6. div-inv 60.0%

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

      7. metadata-eval 60.0%

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

   5. Applied egg-rr 60.0%

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

      1. associate-/l* 61.9%

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

      2. associate-/r/ 62.0%

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

      3. difference-of-squares 67.4%

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

      4. associate-/l* 74.7%

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

      5. *-inverses 74.7%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 74.7%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 74.7%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around inf 75.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 7: 62.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left|angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.2e+129)
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
   (* 0.011111111111111112 (fabs (* angle (* (- b a) (* (+ b a) PI)))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.2e+129) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = 0.011111111111111112 * fabs((angle * ((b - a) * ((b + a) * ((double) M_PI)))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.2e+129) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = 0.011111111111111112 * Math.abs((angle * ((b - a) * ((b + a) * Math.PI))));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 2.2e+129:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = 0.011111111111111112 * math.fabs((angle * ((b - a) * ((b + a) * math.pi))))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.2e+129)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(0.011111111111111112 * abs(Float64(angle * Float64(Float64(b - a) * Float64(Float64(b + a) * pi)))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.2e+129)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = 0.011111111111111112 * abs((angle * ((b - a) * ((b + a) * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 2.2e+129], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[Abs[N[(angle * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.1999999999999999e129

   1. Initial program 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-*l* 62.5%

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

      3. associate-*l* 62.5%

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

      4. unpow2 62.5%

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

      5. unpow2 62.5%

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

      6. difference-of-squares 68.2%

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

   3. Simplified 68.2%

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

      1. associate-*l* 75.9%

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

      2. flip-+ 62.5%

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

      3. associate-*l/ 60.3%

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

      4. associate-*l* 60.3%

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

      5. 2-sin 60.3%

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

      6. div-inv 59.9%

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

      7. metadata-eval 59.9%

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

   5. Applied egg-rr 59.9%

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

      1. associate-/l* 61.9%

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

      2. associate-/r/ 62.0%

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

      3. difference-of-squares 67.3%

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

      4. associate-/l* 74.9%

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

      5. *-inverses 74.9%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 74.9%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 74.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 75.2%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 75.2%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 75.2%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 75.2%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in angle around 0 75.7%

       \[\leadsto \frac{a + b}{1} \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 75.7%

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

   13. Simplified 75.7%

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

   if 2.1999999999999999e129 < angle

   1. Initial program 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-*l* 33.3%

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

      3. associate-*l* 33.3%

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

      4. unpow2 33.3%

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

      5. unpow2 33.3%

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

      6. difference-of-squares 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in angle around 0 16.6%

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

      1. add-sqr-sqrt 13.0%

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

      2. sqrt-unprod 29.5%

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

      3. pow2 29.5%

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

      4. +-commutative 29.5%

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

      5. difference-of-squares 24.5%

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

   6. Applied egg-rr 24.5%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\sqrt{{\left(angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 24.5%

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

      2. rem-sqrt-square 24.6%

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

      3. difference-of-squares 29.6%

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

      4. +-commutative 29.6%

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

      5. associate-*r* 29.6%

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

      6. +-commutative 29.6%

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

   8. Simplified 29.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left|angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right|\\ \end{array} \]

Alternative 8: 62.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right|\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.2e+129)
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
   (* 0.011111111111111112 (* angle (fabs (* (- b a) (* (+ b a) PI)))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.2e+129) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = 0.011111111111111112 * (angle * fabs(((b - a) * ((b + a) * ((double) M_PI)))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.2e+129) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = 0.011111111111111112 * (angle * Math.abs(((b - a) * ((b + a) * Math.PI))));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 2.2e+129:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = 0.011111111111111112 * (angle * math.fabs(((b - a) * ((b + a) * math.pi))))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.2e+129)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * abs(Float64(Float64(b - a) * Float64(Float64(b + a) * pi)))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.2e+129)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = 0.011111111111111112 * (angle * abs(((b - a) * ((b + a) * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 2.2e+129], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[Abs[N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.1999999999999999e129

   1. Initial program 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-*l* 62.5%

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

      3. associate-*l* 62.5%

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

      4. unpow2 62.5%

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

      5. unpow2 62.5%

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

      6. difference-of-squares 68.2%

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

   3. Simplified 68.2%

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

      1. associate-*l* 75.9%

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

      2. flip-+ 62.5%

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

      3. associate-*l/ 60.3%

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

      4. associate-*l* 60.3%

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

      5. 2-sin 60.3%

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

      6. div-inv 59.9%

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

      7. metadata-eval 59.9%

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

   5. Applied egg-rr 59.9%

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

      1. associate-/l* 61.9%

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

      2. associate-/r/ 62.0%

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

      3. difference-of-squares 67.3%

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

      4. associate-/l* 74.9%

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

      5. *-inverses 74.9%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 74.9%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 74.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 75.2%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 75.2%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 75.2%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 75.2%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in angle around 0 75.7%

       \[\leadsto \frac{a + b}{1} \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 75.7%

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

   13. Simplified 75.7%

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

   if 2.1999999999999999e129 < angle

   1. Initial program 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-*l* 33.3%

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

      3. associate-*l* 33.3%

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

      4. unpow2 33.3%

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

      5. unpow2 33.3%

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

      6. difference-of-squares 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in angle around 0 16.6%

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

      1. add-sqr-sqrt 13.0%

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

      2. sqrt-unprod 31.9%

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

      3. pow2 31.9%

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

      4. +-commutative 31.9%

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

      5. difference-of-squares 26.9%

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

   6. Applied egg-rr 26.9%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\sqrt{{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)}^{2}}}\right) \]
   7. Step-by-step derivation

      1. unpow2 26.9%

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

      2. rem-sqrt-square 24.6%

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

      3. difference-of-squares 29.6%

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

      4. +-commutative 29.6%

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

      5. associate-*r* 29.6%

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

      6. +-commutative 29.6%

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

   8. Simplified 29.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left|\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right|\right)\\ \end{array} \]

Alternative 9: 62.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3e+113)
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
   (* (+ b a) (* b (sin (* (* PI angle) 0.011111111111111112))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = (b + a) * (b * sin(((((double) M_PI) * angle) * 0.011111111111111112)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = (b + a) * (b * Math.sin(((Math.PI * angle) * 0.011111111111111112)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 3e+113:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = (b + a) * (b * math.sin(((math.pi * angle) * 0.011111111111111112)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3e+113)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(Float64(pi * angle) * 0.011111111111111112))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 3e+113)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = (b + a) * (b * sin(((pi * angle) * 0.011111111111111112)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 3e+113], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3e113

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

      3. associate-*l* 62.9%

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

      4. unpow2 62.9%

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

      5. unpow2 62.9%

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

      6. difference-of-squares 68.7%

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

   3. Simplified 68.7%

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

      1. associate-*l* 76.5%

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

      2. flip-+ 63.0%

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

      3. associate-*l/ 60.8%

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

      4. associate-*l* 60.8%

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

      5. 2-sin 60.8%

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

      6. div-inv 60.4%

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

      7. metadata-eval 60.4%

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

   5. Applied egg-rr 60.4%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 62.5%

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

      3. difference-of-squares 67.8%

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

      4. associate-/l* 75.5%

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

      5. *-inverses 75.5%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 75.5%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 75.5%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 75.8%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 75.8%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in angle around 0 76.3%

       \[\leadsto \frac{a + b}{1} \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 76.3%

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

   13. Simplified 76.3%

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

   if 3e113 < angle

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-*l* 32.2%

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

      3. associate-*l* 32.2%

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

      4. unpow2 32.2%

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

      5. unpow2 32.2%

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

      6. difference-of-squares 34.6%

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

   3. Simplified 34.6%

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

      1. associate-*l* 34.6%

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

      2. flip-+ 32.2%

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

      3. associate-*l/ 31.5%

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

      4. associate-*l* 31.5%

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

      5. 2-sin 31.5%

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

      6. div-inv 29.5%

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

      7. metadata-eval 29.5%

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

   5. Applied egg-rr 29.5%

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

      1. associate-/l* 30.0%

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

      2. associate-/r/ 30.0%

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

      3. difference-of-squares 32.4%

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

      4. associate-/l* 32.4%

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

      5. *-inverses 32.4%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 32.4%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 32.4%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 33.9%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 33.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 33.0%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 33.0%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in b around inf 31.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 10: 62.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3e+113)
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
   (* (+ b a) (* b (sin (* PI (* angle 0.011111111111111112)))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = (b + a) * (b * sin((((double) M_PI) * (angle * 0.011111111111111112))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = (b + a) * (b * Math.sin((Math.PI * (angle * 0.011111111111111112))));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 3e+113:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = (b + a) * (b * math.sin((math.pi * (angle * 0.011111111111111112))))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3e+113)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(pi * Float64(angle * 0.011111111111111112)))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 3e+113)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = (b + a) * (b * sin((pi * (angle * 0.011111111111111112))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 3e+113], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(b * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3e113

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

      3. associate-*l* 62.9%

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

      4. unpow2 62.9%

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

      5. unpow2 62.9%

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

      6. difference-of-squares 68.7%

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

   3. Simplified 68.7%

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

      1. associate-*l* 76.5%

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

      2. flip-+ 63.0%

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

      3. associate-*l/ 60.8%

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

      4. associate-*l* 60.8%

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

      5. 2-sin 60.8%

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

      6. div-inv 60.4%

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

      7. metadata-eval 60.4%

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

   5. Applied egg-rr 60.4%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 62.5%

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

      3. difference-of-squares 67.8%

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

      4. associate-/l* 75.5%

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

      5. *-inverses 75.5%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 75.5%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 75.5%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 75.8%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 75.8%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in angle around 0 76.3%

       \[\leadsto \frac{a + b}{1} \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 76.3%

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

   13. Simplified 76.3%

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

   if 3e113 < angle

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-*l* 32.2%

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

      3. associate-*l* 32.2%

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

      4. unpow2 32.2%

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

      5. unpow2 32.2%

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

      6. difference-of-squares 34.6%

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

   3. Simplified 34.6%

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

      1. associate-*l* 34.6%

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

      2. flip-+ 32.2%

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

      3. associate-*l/ 31.5%

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

      4. associate-*l* 31.5%

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

      5. 2-sin 31.5%

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

      6. div-inv 29.5%

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

      7. metadata-eval 29.5%

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

   5. Applied egg-rr 29.5%

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

      1. associate-/l* 30.0%

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

      2. associate-/r/ 30.0%

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

      3. difference-of-squares 32.4%

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

      4. associate-/l* 32.4%

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

      5. *-inverses 32.4%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 32.4%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 32.4%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 33.9%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 33.9%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 33.0%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 33.0%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in b around inf 31.5%

       \[\leadsto \frac{a + b}{1} \cdot \color{blue}{\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative 31.5%

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

       2. metadata-eval 31.5%

          \[\leadsto \frac{a + b}{1} \cdot \left(\sin \left(\color{blue}{\left(2 \cdot 0.005555555555555556\right)} \cdot \left(angle \cdot \pi\right)\right) \cdot b\right) \]

       3. associate-*r* 31.5%

          \[\leadsto \frac{a + b}{1} \cdot \left(\sin \color{blue}{\left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot b\right) \]

       4. associate-*r* 31.3%

          \[\leadsto \frac{a + b}{1} \cdot \left(\sin \left(2 \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right) \cdot b\right) \]

       5. *-commutative 31.3%

          \[\leadsto \frac{a + b}{1} \cdot \left(\sin \left(2 \cdot \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \cdot b\right) \]

       6. *-commutative 31.3%

          \[\leadsto \frac{a + b}{1} \cdot \left(\sin \left(2 \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right) \cdot b\right) \]

       7. associate-*r* 31.5%

          \[\leadsto \frac{a + b}{1} \cdot \left(\sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot b\right) \]

       8. associate-*r* 31.5%

          \[\leadsto \frac{a + b}{1} \cdot \left(\sin \color{blue}{\left(\left(2 \cdot 0.005555555555555556\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot b\right) \]

       9. metadata-eval 31.5%

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

       10. associate-*r* 31.3%

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

   13. Simplified 31.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 11: 62.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3e+113)
   (* (+ b a) (* 0.011111111111111112 (* PI (* (- b a) angle))))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * (0.011111111111111112 * (((double) M_PI) * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * (0.011111111111111112 * (Math.PI * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 3e+113:
		tmp = (b + a) * (0.011111111111111112 * (math.pi * ((b - a) * angle)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3e+113)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(b - a) * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 3e+113)
		tmp = (b + a) * (0.011111111111111112 * (pi * ((b - a) * angle)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 3e+113], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3e113

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

      3. associate-*l* 62.9%

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

      4. unpow2 62.9%

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

      5. unpow2 62.9%

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

      6. difference-of-squares 68.7%

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

   3. Simplified 68.7%

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

      1. associate-*l* 76.5%

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

      2. flip-+ 63.0%

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

      3. associate-*l/ 60.8%

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

      4. associate-*l* 60.8%

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

      5. 2-sin 60.8%

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

      6. div-inv 60.4%

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

      7. metadata-eval 60.4%

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

   5. Applied egg-rr 60.4%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 62.5%

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

      3. difference-of-squares 67.8%

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

      4. associate-/l* 75.5%

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

      5. *-inverses 75.5%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 75.5%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 75.5%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 75.8%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 75.8%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in angle around 0 76.3%

       \[\leadsto \frac{a + b}{1} \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative 76.3%

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

       2. associate-*l* 76.3%

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

   13. Simplified 76.3%

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

   if 3e113 < angle

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-*l* 32.2%

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

      3. associate-*l* 32.2%

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

      4. unpow2 32.2%

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

      5. unpow2 32.2%

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

      6. difference-of-squares 34.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in angle around 0 16.1%

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

   5. Taylor expanded in a around 0 23.8%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 23.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

      2. unpow2 23.8%

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

   7. Simplified 23.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 12: 62.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3e+113)
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3e+113) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 3e+113:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3e+113)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 3e+113)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 3e+113], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3e113

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

      3. associate-*l* 62.9%

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

      4. unpow2 62.9%

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

      5. unpow2 62.9%

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

      6. difference-of-squares 68.7%

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

   3. Simplified 68.7%

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

      1. associate-*l* 76.5%

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

      2. flip-+ 63.0%

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

      3. associate-*l/ 60.8%

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

      4. associate-*l* 60.8%

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

      5. 2-sin 60.8%

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

      6. div-inv 60.4%

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

      7. metadata-eval 60.4%

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

   5. Applied egg-rr 60.4%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 62.5%

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

      3. difference-of-squares 67.8%

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

      4. associate-/l* 75.5%

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

      5. *-inverses 75.5%

         \[\leadsto \frac{b + a}{\color{blue}{1}} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      6. +-commutative 75.5%

         \[\leadsto \frac{\color{blue}{a + b}}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \]

      7. *-commutative 75.5%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot angle\right)}\right)\right)\right) \]

   7. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 75.8%

      \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \]

      2. associate-*r* 75.8%

         \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   10. Simplified 75.8%

       \[\leadsto \frac{a + b}{1} \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \]

   11. Taylor expanded in angle around 0 76.3%

       \[\leadsto \frac{a + b}{1} \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 76.3%

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

   13. Simplified 76.3%

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

   if 3e113 < angle

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-*l* 32.2%

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

      3. associate-*l* 32.2%

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

      4. unpow2 32.2%

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

      5. unpow2 32.2%

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

      6. difference-of-squares 34.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in angle around 0 16.1%

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

   5. Taylor expanded in a around 0 23.8%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 23.8%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

      2. unpow2 23.8%

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

   7. Simplified 23.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{+113}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 13: 53.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* PI (* (+ b a) (- b a))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * ((b + a) * (b - a))));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * ((b + a) * (b - a))));
}

Python

b = abs(b)
def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * ((b + a) * (b - a))))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b + a) * Float64(b - a)))))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * ((b + a) * (b - a))));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

   1. *-commutative 57.9%

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

   2. associate-*l* 57.9%

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

   3. associate-*l* 57.9%

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

   4. unpow2 57.9%

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

   5. unpow2 57.9%

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

   6. difference-of-squares 63.1%

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

3. Simplified 63.1%

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

4. Taylor expanded in angle around 0 60.3%

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

5. Final simplification 60.3%

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

Alternative 14: 46.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+61}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.2e+61)
   (* (* PI angle) (* (* a a) -0.011111111111111112))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.2e+61) {
		tmp = (((double) M_PI) * angle) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.2e+61) {
		tmp = (Math.PI * angle) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 5.2e+61:
		tmp = (math.pi * angle) * ((a * a) * -0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.2e+61)
		tmp = Float64(Float64(pi * angle) * Float64(Float64(a * a) * -0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5.2e+61)
		tmp = (pi * angle) * ((a * a) * -0.011111111111111112);
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 5.2e+61], N[(N[(Pi * angle), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.19999999999999945e61

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.3%

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

      3. associate-*l* 59.3%

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

      4. unpow2 59.3%

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

      5. unpow2 59.3%

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

      6. difference-of-squares 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in angle around 0 59.3%

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

   5. Taylor expanded in a around inf 43.7%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-*l* 43.7%

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

      4. unpow2 43.7%

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

   7. Simplified 43.7%

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

   if 5.19999999999999945e61 < b

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*l* 51.3%

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

      3. associate-*l* 51.3%

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

      4. unpow2 51.3%

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

      5. unpow2 51.3%

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

      6. difference-of-squares 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in angle around 0 65.1%

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

   5. Taylor expanded in a around 0 55.3%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 55.3%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

      2. unpow2 55.3%

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

   7. Simplified 55.3%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+61}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 15: 34.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* PI (* b b)))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * (b * b)));
}

Python

b = abs(b)
def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * (b * b)))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

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

   1. *-commutative 57.9%

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

   2. associate-*l* 57.9%

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

   3. associate-*l* 57.9%

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

   4. unpow2 57.9%

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

   5. unpow2 57.9%

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

   6. difference-of-squares 63.1%

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

3. Simplified 63.1%

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

4. Taylor expanded in angle around 0 60.3%

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

5. Taylor expanded in a around 0 40.5%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 40.5%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

   2. unpow2 40.5%

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

7. Simplified 40.5%

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

8. Final simplification 40.5%

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

Reproduce

herbie shell --seed 2023279 
(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)))))