ab-angle->ABCF B

Percentage Accurate: 53.8% → 65.9%

Time: 59.4s

Alternatives: 19

Speedup: 5.4×

• 32.5% 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: 53.8% 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: 65.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-79}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+203}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt[3]{angle} \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot {\left(\sqrt[3]{angle}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.2e-79)
   (* (+ b a) (* (- b a) (sin (* 2.0 (* PI (* angle 0.005555555555555556))))))
   (if (<= b 2.8e+203)
     (*
      (+ b a)
      (*
       (- b a)
       (sin
        (*
         2.0
         (*
          (cbrt angle)
          (* (* PI 0.005555555555555556) (pow (cbrt angle) 2.0)))))))
     (* (+ b a) (* b (sin (* PI (* angle 0.011111111111111112))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.2e-79) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (((double) M_PI) * (angle * 0.005555555555555556)))));
	} else if (b <= 2.8e+203) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (cbrt(angle) * ((((double) M_PI) * 0.005555555555555556) * pow(cbrt(angle), 2.0))))));
	} else {
		tmp = (b + a) * (b * sin((((double) M_PI) * (angle * 0.011111111111111112))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.2e-79) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.PI * (angle * 0.005555555555555556)))));
	} else if (b <= 2.8e+203) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.cbrt(angle) * ((Math.PI * 0.005555555555555556) * Math.pow(Math.cbrt(angle), 2.0))))));
	} else {
		tmp = (b + a) * (b * Math.sin((Math.PI * (angle * 0.011111111111111112))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.2e-79)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * Float64(angle * 0.005555555555555556))))));
	elseif (b <= 2.8e+203)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(cbrt(angle) * Float64(Float64(pi * 0.005555555555555556) * (cbrt(angle) ^ 2.0)))))));
	else
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(pi * Float64(angle * 0.011111111111111112)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 6.2e-79], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e+203], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[Power[angle, 1/3], $MachinePrecision] * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[angle, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 3 regimes

2. if b < 6.1999999999999999e-79

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 61.9%

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

      3. associate-*l* 61.9%

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

      4. unpow2 61.9%

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

      5. unpow2 61.9%

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

      6. difference-of-squares 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in angle around inf 62.8%

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

      1. expm1-log1p-u 49.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 34.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 55.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 72.6%

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

      3. *-commutative 72.6%

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

   8. Simplified 72.6%

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

   if 6.1999999999999999e-79 < b < 2.7999999999999999e203

   1. Initial program 48.1%

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

      1. *-commutative 48.1%

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

      2. associate-*l* 48.1%

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

      3. associate-*l* 48.1%

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

      4. unpow2 48.1%

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

      5. unpow2 48.1%

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

      6. difference-of-squares 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in angle around inf 43.3%

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

      1. expm1-log1p-u 30.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 15.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 17.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 33.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 53.3%

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

      3. *-commutative 53.3%

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

   8. Simplified 53.3%

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

      1. *-commutative 53.3%

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

      2. add-cube-cbrt 52.3%

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

      3. pow3 55.5%

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

      4. *-commutative 55.5%

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

   10. Applied egg-rr 55.5%

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

       1. unpow3 52.3%

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

       2. add-cube-cbrt 53.3%

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

       3. *-commutative 53.3%

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

       4. associate-*r* 55.5%

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

       5. add-cube-cbrt 55.1%

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

       6. associate-*r* 59.4%

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

       7. pow2 59.4%

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

   12. Applied egg-rr 59.4%

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

   if 2.7999999999999999e203 < b

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 61.9%

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

      3. associate-*l* 61.9%

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

      4. unpow2 61.9%

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

      5. unpow2 61.9%

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

      6. difference-of-squares 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in angle around inf 70.6%

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

      1. expm1-log1p-u 35.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 35.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 38.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 38.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 82.5%

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

      3. *-commutative 82.5%

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

   8. Simplified 82.5%

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

   9. Taylor expanded in b around inf 82.5%

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

       1. *-commutative 82.5%

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

       2. *-commutative 82.5%

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

       3. *-commutative 82.5%

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

       4. associate-*l* 86.9%

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

   11. Simplified 86.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-79}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+203}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt[3]{angle} \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot {\left(\sqrt[3]{angle}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 2: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;b \leq 4.5 \cdot 10^{+142}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot t_0\right)\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+211}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{\pi \cdot angle} \cdot \sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{t_0}\right)}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (if (<= b 4.5e+142)
     (* (+ b a) (* (- b a) (sin (* 2.0 t_0))))
     (if (<= b 4e+211)
       (*
        (+ b a)
        (*
         (- b a)
         (sin
          (*
           2.0
           (pow (* (cbrt (* PI angle)) (cbrt 0.005555555555555556)) 3.0)))))
       (* (+ b a) (* (- b a) (sin (* 2.0 (pow (cbrt t_0) 3.0)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if (b <= 4.5e+142) {
		tmp = (b + a) * ((b - a) * sin((2.0 * t_0)));
	} else if (b <= 4e+211) {
		tmp = (b + a) * ((b - a) * sin((2.0 * pow((cbrt((((double) M_PI) * angle)) * cbrt(0.005555555555555556)), 3.0))));
	} else {
		tmp = (b + a) * ((b - a) * sin((2.0 * pow(cbrt(t_0), 3.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if (b <= 4.5e+142) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * t_0)));
	} else if (b <= 4e+211) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * Math.pow((Math.cbrt((Math.PI * angle)) * Math.cbrt(0.005555555555555556)), 3.0))));
	} else {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * Math.pow(Math.cbrt(t_0), 3.0))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (b <= 4.5e+142)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * t_0))));
	elseif (b <= 4e+211)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * (Float64(cbrt(Float64(pi * angle)) * cbrt(0.005555555555555556)) ^ 3.0)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * (cbrt(t_0) ^ 3.0)))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.5e+142], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+211], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Power[N[(N[Power[N[(Pi * angle), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[0.005555555555555556, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.4999999999999999e142

   1. Initial program 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-*l* 59.2%

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

      3. associate-*l* 59.2%

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

      4. unpow2 59.2%

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

      5. unpow2 59.2%

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

      6. difference-of-squares 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in angle around inf 58.9%

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

      1. expm1-log1p-u 46.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 30.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 50.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 67.9%

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

      3. *-commutative 67.9%

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

   8. Simplified 67.9%

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

   if 4.4999999999999999e142 < b < 3.9999999999999998e211

   1. Initial program 23.0%

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

      1. *-commutative 23.0%

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

      2. associate-*l* 23.0%

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

      3. associate-*l* 23.0%

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

      4. unpow2 23.0%

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

      5. unpow2 23.0%

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

      6. difference-of-squares 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in angle around inf 13.0%

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

      1. expm1-log1p-u 0.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 0.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 0.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 0.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 49.8%

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

      3. *-commutative 49.8%

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

   8. Simplified 49.8%

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

      1. *-commutative 49.8%

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

      2. add-cube-cbrt 49.0%

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

      3. pow3 58.8%

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

      4. *-commutative 58.8%

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

   10. Applied egg-rr 58.8%

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

       1. associate-*r* 58.7%

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

       2. *-commutative 58.7%

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

       3. cbrt-prod 78.5%

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

       4. *-commutative 78.5%

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

   12. Applied egg-rr 78.5%

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

   if 3.9999999999999998e211 < b

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

      2. associate-*l* 67.5%

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

      3. associate-*l* 67.5%

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

      4. unpow2 67.5%

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

      5. unpow2 67.5%

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

      6. difference-of-squares 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in angle around inf 77.0%

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

      1. expm1-log1p-u 38.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 38.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 42.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 42.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 85.6%

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

      3. *-commutative 85.6%

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

   8. Simplified 85.6%

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

      1. *-commutative 85.6%

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

      2. add-cube-cbrt 71.4%

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

      3. pow3 71.4%

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

      4. *-commutative 71.4%

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

   10. Applied egg-rr 71.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{+142}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+211}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{\pi \cdot angle} \cdot \sqrt[3]{0.005555555555555556}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\\ \end{array} \]

Alternative 3: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{\pi}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 2e+101)
   (*
    (+ b a)
    (*
     (- b a)
     (sin (* 2.0 (pow (cbrt (* angle (* PI 0.005555555555555556))) 3.0)))))
   (*
    (+ b a)
    (*
     (- b a)
     (sin
      (* 2.0 (* (sqrt PI) (* (* angle 0.005555555555555556) (sqrt PI)))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+101) {
		tmp = (b + a) * ((b - a) * sin((2.0 * pow(cbrt((angle * (((double) M_PI) * 0.005555555555555556))), 3.0))));
	} else {
		tmp = (b + a) * ((b - a) * sin((2.0 * (sqrt(((double) M_PI)) * ((angle * 0.005555555555555556) * sqrt(((double) M_PI)))))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 2e+101) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * Math.pow(Math.cbrt((angle * (Math.PI * 0.005555555555555556))), 3.0))));
	} else {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.sqrt(Math.PI) * ((angle * 0.005555555555555556) * Math.sqrt(Math.PI))))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 2e+101)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * (cbrt(Float64(angle * Float64(pi * 0.005555555555555556))) ^ 3.0)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(sqrt(pi) * Float64(Float64(angle * 0.005555555555555556) * sqrt(pi)))))));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+101], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Power[N[Power[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 2e101

   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 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in angle around inf 63.3%

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

      1. expm1-log1p-u 48.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 31.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 53.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 74.9%

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

      3. *-commutative 74.9%

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

   8. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cube-cbrt 74.0%

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

      3. pow3 76.0%

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

      4. *-commutative 76.0%

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

   10. Applied egg-rr 76.0%

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

       1. expm1-log1p-u 64.9%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)}}^{3}\right)\right) \]

       2. expm1-udef 25.4%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)} - 1\right)}}^{3}\right)\right) \]

   12. Applied egg-rr 25.4%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)} - 1\right)}}^{3}\right)\right) \]
   13. Step-by-step derivation

       1. expm1-def 64.9%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\pi \cdot \left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)}}^{3}\right)\right) \]

       2. expm1-log1p 76.0%

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

       3. *-commutative 76.0%

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

       4. associate-*l* 76.1%

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

       5. *-commutative 76.1%

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

   14. Simplified 76.1%

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

   if 2e101 < (/.f64 angle 180)

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-*l* 39.1%

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

      3. associate-*l* 39.1%

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

      4. unpow2 39.1%

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

      5. unpow2 39.1%

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

      6. difference-of-squares 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in angle around inf 35.9%

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

      1. expm1-log1p-u 21.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 20.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 23.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 38.5%

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

      3. *-commutative 38.5%

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

   8. Simplified 38.5%

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

      1. *-commutative 38.5%

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

      2. add-cube-cbrt 35.4%

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

      3. pow3 38.7%

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

      4. *-commutative 38.7%

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

   10. Applied egg-rr 38.7%

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

       1. unpow3 35.4%

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

       2. add-cube-cbrt 38.5%

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

       3. *-commutative 38.5%

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

       4. *-commutative 38.5%

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

       5. add-sqr-sqrt 46.2%

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

       6. associate-*r* 48.3%

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

       7. *-commutative 48.3%

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

   12. Applied egg-rr 48.3%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot {\left(\sqrt[3]{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\sqrt{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{\pi}\right)\right)\right)\right)\\ \end{array} \]

Alternative 4: 66.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sqrt{{t_0}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.011111111111111112 (* PI angle)))))
   (if (<= (/ angle 180.0) -2e+149)
     (*
      (+ b a)
      (* (- b a) (sin (* 2.0 (fabs (* angle (* PI 0.005555555555555556)))))))
     (if (<= (/ angle 180.0) 5e+192)
       (* (+ b a) (* (- b a) t_0))
       (* (+ b a) (* (- b a) (sqrt (pow t_0 2.0))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = sin((0.011111111111111112 * (((double) M_PI) * angle)));
	double tmp;
	if ((angle / 180.0) <= -2e+149) {
		tmp = (b + a) * ((b - a) * sin((2.0 * fabs((angle * (((double) M_PI) * 0.005555555555555556))))));
	} else if ((angle / 180.0) <= 5e+192) {
		tmp = (b + a) * ((b - a) * t_0);
	} else {
		tmp = (b + a) * ((b - a) * sqrt(pow(t_0, 2.0)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.sin((0.011111111111111112 * (Math.PI * angle)));
	double tmp;
	if ((angle / 180.0) <= -2e+149) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * Math.abs((angle * (Math.PI * 0.005555555555555556))))));
	} else if ((angle / 180.0) <= 5e+192) {
		tmp = (b + a) * ((b - a) * t_0);
	} else {
		tmp = (b + a) * ((b - a) * Math.sqrt(Math.pow(t_0, 2.0)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.sin((0.011111111111111112 * (math.pi * angle)))
	tmp = 0
	if (angle / 180.0) <= -2e+149:
		tmp = (b + a) * ((b - a) * math.sin((2.0 * math.fabs((angle * (math.pi * 0.005555555555555556))))))
	elif (angle / 180.0) <= 5e+192:
		tmp = (b + a) * ((b - a) * t_0)
	else:
		tmp = (b + a) * ((b - a) * math.sqrt(math.pow(t_0, 2.0)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = sin(Float64(0.011111111111111112 * Float64(pi * angle)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+149)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * abs(Float64(angle * Float64(pi * 0.005555555555555556)))))));
	elseif (Float64(angle / 180.0) <= 5e+192)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * t_0));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sqrt((t_0 ^ 2.0))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = sin((0.011111111111111112 * (pi * angle)));
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+149)
		tmp = (b + a) * ((b - a) * sin((2.0 * abs((angle * (pi * 0.005555555555555556))))));
	elseif ((angle / 180.0) <= 5e+192)
		tmp = (b + a) * ((b - a) * t_0);
	else
		tmp = (b + a) * ((b - a) * sqrt((t_0 ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+149], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Abs[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e+192], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e149

   1. Initial program 16.0%

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

      1. *-commutative 16.0%

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

      2. associate-*l* 16.0%

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

      3. associate-*l* 16.0%

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

      4. unpow2 16.0%

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

      5. unpow2 16.0%

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

      6. difference-of-squares 24.0%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in angle around inf 14.2%

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

      1. expm1-log1p-u 8.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 5.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 12.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 19.6%

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

      3. *-commutative 19.6%

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

   8. Simplified 19.6%

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

      1. *-commutative 19.6%

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

      2. add-cube-cbrt 19.3%

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

      3. pow3 30.7%

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

      4. *-commutative 30.7%

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

   10. Applied egg-rr 30.7%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 4.1%

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

       3. unpow3 4.1%

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

       4. add-cube-cbrt 4.7%

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

       5. associate-*r* 4.7%

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

       6. *-commutative 4.7%

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

       7. *-commutative 4.7%

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

       8. unpow3 4.7%

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

       9. add-cube-cbrt 4.6%

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

       10. associate-*r* 8.6%

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

       11. *-commutative 8.6%

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

       12. *-commutative 8.6%

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

       13. associate-*r* 4.6%

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

       14. *-commutative 4.6%

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

       15. associate-*r* 4.6%

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

       16. *-commutative 4.6%

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

   12. Applied egg-rr 4.0%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\sqrt{{\left(\pi \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 4.0%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \sqrt{\color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)} \cdot 3.08641975308642 \cdot 10^{-5}}\right)\right) \]

       2. metadata-eval 4.0%

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

       3. swap-sqr 8.6%

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

       4. associate-*r* 4.6%

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

       5. associate-*r* 4.6%

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

       6. rem-sqrt-square 32.9%

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

       7. *-commutative 32.9%

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

       8. associate-*l* 36.6%

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

       9. *-commutative 36.6%

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

   14. Simplified 36.6%

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

   if -2.0000000000000001e149 < (/.f64 angle 180) < 5.00000000000000033e192

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

      2. associate-*l* 67.2%

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

      3. associate-*l* 67.2%

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

      4. unpow2 67.2%

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

      5. unpow2 67.2%

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

      6. difference-of-squares 69.2%

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

   3. Simplified 69.2%

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

      1. difference-of-squares 67.2%

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

      2. *-commutative 67.2%

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

      3. sub-neg 67.2%

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

      4. distribute-lft-in 67.2%

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

      5. associate-*l* 67.2%

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

      6. 2-sin 67.2%

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

      7. div-inv 66.6%

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

      8. metadata-eval 66.6%

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

   5. Applied egg-rr 66.5%

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

      1. distribute-lft-out 66.5%

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

      2. sub-neg 66.5%

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

      3. difference-of-squares 68.5%

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

      4. *-commutative 68.5%

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

      5. associate-*l* 79.5%

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

      6. +-commutative 79.5%

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

      7. *-commutative 79.5%

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

      8. associate-*r* 80.2%

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

      9. associate-*l* 80.2%

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

      10. *-commutative 80.2%

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

      11. metadata-eval 80.2%

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

   7. Simplified 80.2%

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

   if 5.00000000000000033e192 < (/.f64 angle 180)

   1. Initial program 27.2%

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

      1. *-commutative 27.2%

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

      2. associate-*l* 27.2%

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

      3. associate-*l* 27.2%

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

      4. unpow2 27.2%

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

      5. unpow2 27.2%

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

      6. difference-of-squares 27.2%

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

   3. Simplified 27.2%

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

   4. Taylor expanded in angle around inf 28.6%

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

      1. expm1-log1p-u 20.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 18.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 17.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 18.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 26.8%

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

      3. *-commutative 26.8%

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

   8. Simplified 26.8%

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

      1. add-sqr-sqrt 11.6%

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

      2. sqrt-unprod 43.7%

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

      3. pow2 43.7%

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

      4. *-commutative 43.7%

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

      5. associate-*r* 44.6%

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

      6. *-commutative 44.6%

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

      7. associate-*l* 44.6%

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

      8. *-commutative 44.6%

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

      9. metadata-eval 44.6%

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

   10. Applied egg-rr 44.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sqrt{{\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)}^{2}}\right)\\ \end{array} \]

Alternative 5: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+140}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -2e+149)
   (*
    (+ b a)
    (* (- b a) (sin (* 2.0 (fabs (* angle (* PI 0.005555555555555556)))))))
   (if (<= (/ angle 180.0) 1e+140)
     (* (+ b a) (* (- b a) (sin (* 0.011111111111111112 (* PI angle)))))
     (* (* (+ b a) (- b a)) (* 2.0 (sin (* PI (/ angle 180.0))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+149) {
		tmp = (b + a) * ((b - a) * sin((2.0 * fabs((angle * (((double) M_PI) * 0.005555555555555556))))));
	} else if ((angle / 180.0) <= 1e+140) {
		tmp = (b + a) * ((b - a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = ((b + a) * (b - a)) * (2.0 * sin((((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -2e+149) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * Math.abs((angle * (Math.PI * 0.005555555555555556))))));
	} else if ((angle / 180.0) <= 1e+140) {
		tmp = (b + a) * ((b - a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else {
		tmp = ((b + a) * (b - a)) * (2.0 * Math.sin((Math.PI * (angle / 180.0))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= -2e+149:
		tmp = (b + a) * ((b - a) * math.sin((2.0 * math.fabs((angle * (math.pi * 0.005555555555555556))))))
	elif (angle / 180.0) <= 1e+140:
		tmp = (b + a) * ((b - a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	else:
		tmp = ((b + a) * (b - a)) * (2.0 * math.sin((math.pi * (angle / 180.0))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+149)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * abs(Float64(angle * Float64(pi * 0.005555555555555556)))))));
	elseif (Float64(angle / 180.0) <= 1e+140)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(Float64(Float64(b + a) * Float64(b - a)) * Float64(2.0 * sin(Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((angle / 180.0) <= -2e+149)
		tmp = (b + a) * ((b - a) * sin((2.0 * abs((angle * (pi * 0.005555555555555556))))));
	elseif ((angle / 180.0) <= 1e+140)
		tmp = (b + a) * ((b - a) * sin((0.011111111111111112 * (pi * angle))));
	else
		tmp = ((b + a) * (b - a)) * (2.0 * sin((pi * (angle / 180.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+149], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[Abs[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e+140], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e149

   1. Initial program 16.0%

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

      1. *-commutative 16.0%

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

      2. associate-*l* 16.0%

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

      3. associate-*l* 16.0%

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

      4. unpow2 16.0%

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

      5. unpow2 16.0%

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

      6. difference-of-squares 24.0%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in angle around inf 14.2%

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

      1. expm1-log1p-u 8.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 5.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 12.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 19.6%

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

      3. *-commutative 19.6%

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

   8. Simplified 19.6%

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

      1. *-commutative 19.6%

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

      2. add-cube-cbrt 19.3%

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

      3. pow3 30.7%

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

      4. *-commutative 30.7%

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

   10. Applied egg-rr 30.7%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 4.1%

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

       3. unpow3 4.1%

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

       4. add-cube-cbrt 4.7%

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

       5. associate-*r* 4.7%

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

       6. *-commutative 4.7%

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

       7. *-commutative 4.7%

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

       8. unpow3 4.7%

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

       9. add-cube-cbrt 4.6%

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

       10. associate-*r* 8.6%

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

       11. *-commutative 8.6%

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

       12. *-commutative 8.6%

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

       13. associate-*r* 4.6%

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

       14. *-commutative 4.6%

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

       15. associate-*r* 4.6%

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

       16. *-commutative 4.6%

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

   12. Applied egg-rr 4.0%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \color{blue}{\sqrt{{\left(\pi \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}}\right)\right) \]
   13. Step-by-step derivation

       1. unpow2 4.0%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \sqrt{\color{blue}{\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot angle\right)\right)} \cdot 3.08641975308642 \cdot 10^{-5}}\right)\right) \]

       2. metadata-eval 4.0%

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

       3. swap-sqr 8.6%

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

       4. associate-*r* 4.6%

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

       5. associate-*r* 4.6%

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

       6. rem-sqrt-square 32.9%

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

       7. *-commutative 32.9%

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

       8. associate-*l* 36.6%

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

       9. *-commutative 36.6%

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

   14. Simplified 36.6%

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

   if -2.0000000000000001e149 < (/.f64 angle 180) < 1.00000000000000006e140

   1. Initial program 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-*l* 67.8%

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

      3. associate-*l* 67.8%

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

      4. unpow2 67.8%

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

      5. unpow2 67.8%

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

      6. difference-of-squares 69.8%

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

   3. Simplified 69.8%

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

      1. difference-of-squares 67.8%

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

      2. *-commutative 67.8%

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

      3. sub-neg 67.8%

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

      4. distribute-lft-in 67.8%

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

      5. associate-*l* 67.8%

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

      6. 2-sin 67.8%

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

      7. div-inv 67.2%

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

      8. metadata-eval 67.2%

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

   5. Applied egg-rr 67.2%

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

      1. distribute-lft-out 67.2%

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

      2. sub-neg 67.2%

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

      3. difference-of-squares 69.2%

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

      4. *-commutative 69.2%

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

      5. associate-*l* 80.6%

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

      6. +-commutative 80.6%

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

      7. *-commutative 80.6%

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

      8. associate-*r* 81.7%

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

      9. associate-*l* 81.7%

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

      10. *-commutative 81.7%

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

      11. metadata-eval 81.7%

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

   7. Simplified 81.7%

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

   if 1.00000000000000006e140 < (/.f64 angle 180)

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*l* 33.6%

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

      4. unpow2 33.6%

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

      5. unpow2 33.6%

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

      6. difference-of-squares 33.6%

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

   3. Simplified 33.6%

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

   4. Taylor expanded in angle around 0 43.2%

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{1}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left|angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right|\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+140}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \]

Alternative 6: 66.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 5e+212)
   (* (+ b a) (* (- b a) (sin (* 0.011111111111111112 (* PI angle)))))
   (* 2.0 (* (- b a) (* a (sin (* 0.005555555555555556 (* PI angle))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5e+212) {
		tmp = (b + a) * ((b - a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = 2.0 * ((b - a) * (a * sin((0.005555555555555556 * (((double) M_PI) * angle)))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5e+212) {
		tmp = (b + a) * ((b - a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else {
		tmp = 2.0 * ((b - a) * (a * Math.sin((0.005555555555555556 * (Math.PI * angle)))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 5e+212:
		tmp = (b + a) * ((b - a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	else:
		tmp = 2.0 * ((b - a) * (a * math.sin((0.005555555555555556 * (math.pi * angle)))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 5e+212)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(a * sin(Float64(0.005555555555555556 * Float64(pi * angle))))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5e+212)
		tmp = (b + a) * ((b - a) * sin((0.011111111111111112 * (pi * angle))));
	else
		tmp = 2.0 * ((b - a) * (a * sin((0.005555555555555556 * (pi * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 5e+212], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(a * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999992e212

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*l* 58.1%

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

      3. associate-*l* 58.1%

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

      4. unpow2 58.1%

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

      5. unpow2 58.1%

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

      6. difference-of-squares 60.5%

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

   3. Simplified 60.5%

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

      1. difference-of-squares 58.1%

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

      2. *-commutative 58.1%

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

      3. sub-neg 58.1%

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

      4. distribute-lft-in 58.1%

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

      5. associate-*l* 58.1%

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

      6. 2-sin 58.1%

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

      7. div-inv 57.5%

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

      8. metadata-eval 57.5%

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

   5. Applied egg-rr 57.4%

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

      1. distribute-lft-out 57.4%

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

      2. sub-neg 57.4%

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

      3. difference-of-squares 59.5%

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

      4. *-commutative 59.5%

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

      5. associate-*l* 68.1%

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

      6. +-commutative 68.1%

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

      7. *-commutative 68.1%

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

      8. associate-*r* 67.8%

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

      9. associate-*l* 67.8%

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

      10. *-commutative 67.8%

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

      11. metadata-eval 67.8%

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

   7. Simplified 67.8%

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

   if 4.99999999999999992e212 < a

   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. Taylor expanded in angle around inf 78.5%

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

      1. associate-*r* 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*r* 78.5%

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

      4. *-commutative 78.5%

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

      5. *-commutative 78.5%

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

      6. *-commutative 78.5%

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

      7. associate-*r* 78.5%

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

      8. *-commutative 78.5%

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

      9. *-commutative 78.5%

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

      10. *-commutative 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in a around inf 78.5%

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

   8. Taylor expanded in angle around 0 78.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 61.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 7.2 \cdot 10^{+94}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{elif}\;angle \leq 3.8 \cdot 10^{+201}:\\ \;\;\;\;\left|\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 7.2e+94)
   (* (+ b a) (* 0.011111111111111112 (* PI (* (- b a) angle))))
   (if (<= angle 3.8e+201)
     (fabs (* PI (* -0.011111111111111112 (* angle (* a a)))))
     (* 0.011111111111111112 (* angle (* (- b a) (* b PI)))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 7.2e+94) {
		tmp = (b + a) * (0.011111111111111112 * (((double) M_PI) * ((b - a) * angle)));
	} else if (angle <= 3.8e+201) {
		tmp = fabs((((double) M_PI) * (-0.011111111111111112 * (angle * (a * a)))));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 7.2e+94) {
		tmp = (b + a) * (0.011111111111111112 * (Math.PI * ((b - a) * angle)));
	} else if (angle <= 3.8e+201) {
		tmp = Math.abs((Math.PI * (-0.011111111111111112 * (angle * (a * a)))));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * Math.PI)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 7.2e+94:
		tmp = (b + a) * (0.011111111111111112 * (math.pi * ((b - a) * angle)))
	elif angle <= 3.8e+201:
		tmp = math.fabs((math.pi * (-0.011111111111111112 * (angle * (a * a)))))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * math.pi)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 7.2e+94)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(b - a) * angle))));
	elseif (angle <= 3.8e+201)
		tmp = abs(Float64(pi * Float64(-0.011111111111111112 * Float64(angle * Float64(a * a)))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(b * pi))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 7.2e+94)
		tmp = (b + a) * (0.011111111111111112 * (pi * ((b - a) * angle)));
	elseif (angle <= 3.8e+201)
		tmp = abs((pi * (-0.011111111111111112 * (angle * (a * a)))));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 7.2e+94], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 3.8e+201], N[Abs[N[(Pi * N[(-0.011111111111111112 * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 7.19999999999999985e94

   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 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in angle around inf 63.3%

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

      1. expm1-log1p-u 48.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 31.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 53.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 75.1%

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

      3. *-commutative 75.1%

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

   8. Simplified 75.1%

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

   9. Taylor expanded in angle around 0 73.6%

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

       1. associate-*r* 73.6%

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

       2. *-commutative 73.6%

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

   11. Simplified 73.6%

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

   if 7.19999999999999985e94 < angle < 3.79999999999999995e201

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-*l* 51.7%

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

      3. associate-*l* 51.7%

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

      4. unpow2 51.7%

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

      5. unpow2 51.7%

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

      6. difference-of-squares 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in angle around 0 23.2%

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

   5. Taylor expanded in b around 0 19.3%

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

      1. *-commutative 19.3%

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

      2. *-commutative 19.3%

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

      3. unpow2 19.3%

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

   7. Simplified 19.3%

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

      1. add-sqr-sqrt 13.7%

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

      2. sqrt-unprod 49.2%

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

      3. *-commutative 49.2%

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

      4. *-commutative 49.2%

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

      5. swap-sqr 49.2%

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

      6. metadata-eval 49.2%

         \[\leadsto \sqrt{\color{blue}{0.0001234567901234568} \cdot \left(\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\right)} \]

      7. pow2 49.2%

         \[\leadsto \sqrt{0.0001234567901234568 \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)}^{2}}} \]

      8. *-commutative 49.2%

         \[\leadsto \sqrt{0.0001234567901234568 \cdot {\color{blue}{\left(\left(\pi \cdot \left(a \cdot a\right)\right) \cdot angle\right)}}^{2}} \]

      9. associate-*l* 49.2%

         \[\leadsto \sqrt{0.0001234567901234568 \cdot {\color{blue}{\left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)}}^{2}} \]

   9. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{\sqrt{0.0001234567901234568 \cdot {\left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. *-commutative 49.2%

          \[\leadsto \sqrt{\color{blue}{{\left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)}^{2} \cdot 0.0001234567901234568}} \]

       2. unpow2 49.2%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)\right)} \cdot 0.0001234567901234568} \]

       3. metadata-eval 49.2%

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

       4. swap-sqr 49.2%

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

       5. rem-sqrt-square 45.2%

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

       6. associate-*l* 45.2%

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

       7. *-commutative 45.2%

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

       8. unpow2 45.2%

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

       9. *-commutative 45.2%

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

       10. unpow2 45.2%

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

   11. Simplified 45.2%

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

   if 3.79999999999999995e201 < angle

   1. Initial program 28.4%

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

      1. *-commutative 28.4%

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

      2. associate-*l* 28.4%

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

      3. associate-*l* 28.4%

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

      4. unpow2 28.4%

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

      5. unpow2 28.4%

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

      6. difference-of-squares 28.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in angle around 0 40.8%

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

   5. Taylor expanded in a around 0 40.4%

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

      1. *-commutative 40.4%

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

   7. Simplified 40.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 7.2 \cdot 10^{+94}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{elif}\;angle \leq 3.8 \cdot 10^{+201}:\\ \;\;\;\;\left|\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 8: 65.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{-125}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 7e-125)
   (* (+ b a) (* b (sin (* PI (* angle 0.011111111111111112)))))
   (* (+ b a) (* 0.011111111111111112 (* PI (* (- b a) angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7e-125) {
		tmp = (b + a) * (b * sin((((double) M_PI) * (angle * 0.011111111111111112))));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (((double) M_PI) * ((b - a) * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7e-125) {
		tmp = (b + a) * (b * Math.sin((Math.PI * (angle * 0.011111111111111112))));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (Math.PI * ((b - a) * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 7e-125:
		tmp = (b + a) * (b * math.sin((math.pi * (angle * 0.011111111111111112))))
	else:
		tmp = (b + a) * (0.011111111111111112 * (math.pi * ((b - a) * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 7e-125)
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(pi * Float64(angle * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(b - a) * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 7e-125)
		tmp = (b + a) * (b * sin((pi * (angle * 0.011111111111111112))));
	else
		tmp = (b + a) * (0.011111111111111112 * (pi * ((b - a) * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 7e-125], N[(N[(b + a), $MachinePrecision] * N[(b * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.99999999999999995e-125

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 58.9%

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

      3. associate-*l* 58.9%

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

      4. unpow2 58.9%

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

      5. unpow2 58.9%

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

      6. difference-of-squares 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in angle around inf 57.9%

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

      1. expm1-log1p-u 43.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 31.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 49.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 67.9%

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

      3. *-commutative 67.9%

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

   8. Simplified 67.9%

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

   9. Taylor expanded in b around inf 52.0%

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

       1. *-commutative 52.0%

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

       2. *-commutative 52.0%

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

       3. *-commutative 52.0%

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

       4. associate-*l* 51.3%

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

   11. Simplified 51.3%

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

   if 6.99999999999999995e-125 < a

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

      2. associate-*l* 57.5%

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

      3. associate-*l* 57.5%

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

      4. unpow2 57.5%

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

      5. unpow2 57.5%

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

      6. difference-of-squares 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in angle around inf 59.8%

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

      1. expm1-log1p-u 42.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 26.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 46.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 70.2%

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

      3. *-commutative 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in angle around 0 67.3%

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

       1. associate-*r* 67.2%

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

       2. *-commutative 67.2%

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

   11. Simplified 67.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{-125}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 9: 65.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 7e-101)
   (* (+ b a) (* b (sin (* 0.011111111111111112 (* PI angle)))))
   (* (+ b a) (* 0.011111111111111112 (* PI (* (- b a) angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7e-101) {
		tmp = (b + a) * (b * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (((double) M_PI) * ((b - a) * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7e-101) {
		tmp = (b + a) * (b * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (Math.PI * ((b - a) * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 7e-101:
		tmp = (b + a) * (b * math.sin((0.011111111111111112 * (math.pi * angle))))
	else:
		tmp = (b + a) * (0.011111111111111112 * (math.pi * ((b - a) * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 7e-101)
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(b - a) * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 7e-101)
		tmp = (b + a) * (b * sin((0.011111111111111112 * (pi * angle))));
	else
		tmp = (b + a) * (0.011111111111111112 * (pi * ((b - a) * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 7e-101], N[(N[(b + a), $MachinePrecision] * N[(b * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(Pi * N[(N[(b - a), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.99999999999999989e-101

   1. Initial program 58.8%

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

      1. *-commutative 58.8%

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

      2. associate-*l* 58.8%

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

      3. associate-*l* 58.8%

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

      4. unpow2 58.8%

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

      5. unpow2 58.8%

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

      6. difference-of-squares 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in angle around inf 57.8%

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

      1. expm1-log1p-u 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 31.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 49.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 67.7%

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

      3. *-commutative 67.7%

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

   8. Simplified 67.7%

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

   9. Taylor expanded in b around inf 52.6%

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

   if 6.99999999999999989e-101 < a

   1. Initial program 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-*l* 57.7%

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

      3. associate-*l* 57.7%

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

      4. unpow2 57.7%

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

      5. unpow2 57.7%

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

      6. difference-of-squares 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in angle around inf 60.1%

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

      1. expm1-log1p-u 42.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 26.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 46.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 70.6%

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

      3. *-commutative 70.6%

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

   8. Simplified 70.6%

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

   9. Taylor expanded in angle around 0 66.5%

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

       1. associate-*r* 66.5%

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

       2. *-commutative 66.5%

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

   11. Simplified 66.5%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 10: 67.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.5%

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

   1. *-commutative 58.5%

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

   2. associate-*l* 58.5%

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

   3. associate-*l* 58.5%

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

   4. unpow2 58.5%

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

   5. unpow2 58.5%

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

   6. difference-of-squares 60.8%

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

3. Simplified 60.8%

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

   1. difference-of-squares 58.5%

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

   2. *-commutative 58.5%

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

   3. sub-neg 58.5%

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

   4. distribute-lft-in 58.5%

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

   5. associate-*l* 58.5%

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

   6. 2-sin 58.5%

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

   7. div-inv 57.9%

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

   8. metadata-eval 57.9%

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

5. Applied egg-rr 57.8%

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

   1. distribute-lft-out 57.8%

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

   2. sub-neg 57.8%

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

   3. difference-of-squares 59.8%

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

   4. *-commutative 59.8%

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

   5. associate-*l* 68.7%

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

   6. +-commutative 68.7%

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

   7. *-commutative 68.7%

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

   8. associate-*r* 68.4%

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

   9. associate-*l* 68.4%

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

   10. *-commutative 68.4%

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

   11. metadata-eval 68.4%

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

7. Simplified 68.4%

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

8. Final simplification 68.4%

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

Alternative 11: 54.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 920:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 920.0)
   (* 0.011111111111111112 (* angle (* PI (* b b))))
   (if (<= a 5.8e+159)
     (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))
     (* PI (* -0.011111111111111112 (* a (* a angle)))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 920.0) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	} else if (a <= 5.8e+159) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	} else {
		tmp = ((double) M_PI) * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 920.0) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	} else if (a <= 5.8e+159) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	} else {
		tmp = Math.PI * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 920.0:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	elif a <= 5.8e+159:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	else:
		tmp = math.pi * (-0.011111111111111112 * (a * (a * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 920.0)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	elseif (a <= 5.8e+159)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	else
		tmp = Float64(pi * Float64(-0.011111111111111112 * Float64(a * Float64(a * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 920.0)
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	elseif (a <= 5.8e+159)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	else
		tmp = pi * (-0.011111111111111112 * (a * (a * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 920.0], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e+159], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(-0.011111111111111112 * N[(a * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 920

   1. Initial program 59.7%

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

      1. *-commutative 59.7%

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

      2. associate-*l* 59.7%

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

      3. associate-*l* 59.7%

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

      4. unpow2 59.7%

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

      5. unpow2 59.7%

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

      6. difference-of-squares 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in angle around 0 58.5%

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

   5. Taylor expanded in b around inf 44.0%

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

      1. *-commutative 44.0%

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

      2. *-commutative 44.0%

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

      3. unpow2 44.0%

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

   7. Simplified 44.0%

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

   if 920 < a < 5.80000000000000029e159

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

      3. associate-*l* 55.9%

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

      4. unpow2 55.9%

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

      5. unpow2 55.9%

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

      6. difference-of-squares 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in angle around 0 53.4%

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

   5. Taylor expanded in a around inf 38.4%

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

      1. *-commutative 38.4%

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

   7. Simplified 38.4%

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

   if 5.80000000000000029e159 < a

   1. Initial program 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*l* 51.5%

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

      3. associate-*l* 51.5%

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

      4. unpow2 51.5%

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

      5. unpow2 51.5%

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

      6. difference-of-squares 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in angle around 0 56.1%

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

   5. Taylor expanded in b around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. *-commutative 56.1%

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

      3. unpow2 56.1%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in angle around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. associate-*r* 56.1%

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

      3. *-commutative 56.1%

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

      4. unpow2 56.1%

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

      5. *-commutative 56.1%

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

      6. associate-*l* 56.1%

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

      7. *-commutative 56.1%

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

      8. unpow2 56.1%

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

      9. *-commutative 56.1%

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

      10. unpow2 56.1%

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

   10. Simplified 56.1%

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

   11. Taylor expanded in angle around 0 56.1%

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

       1. unpow2 56.1%

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

       2. associate-*r* 77.1%

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

   13. Simplified 77.1%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 920:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 12: 54.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 800:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 800.0)
   (* 0.011111111111111112 (* angle (* (- b a) (* b PI))))
   (if (<= a 5.8e+159)
     (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))
     (* PI (* -0.011111111111111112 (* a (* a angle)))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 800.0) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * ((double) M_PI))));
	} else if (a <= 5.8e+159) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	} else {
		tmp = ((double) M_PI) * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 800.0) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * Math.PI)));
	} else if (a <= 5.8e+159) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	} else {
		tmp = Math.PI * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 800.0:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * math.pi)))
	elif a <= 5.8e+159:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	else:
		tmp = math.pi * (-0.011111111111111112 * (a * (a * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 800.0)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(b * pi))));
	elseif (a <= 5.8e+159)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	else
		tmp = Float64(pi * Float64(-0.011111111111111112 * Float64(a * Float64(a * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 800.0)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (b * pi)));
	elseif (a <= 5.8e+159)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	else
		tmp = pi * (-0.011111111111111112 * (a * (a * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 800.0], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e+159], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(-0.011111111111111112 * N[(a * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 800

   1. Initial program 59.7%

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

      1. *-commutative 59.7%

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

      2. associate-*l* 59.7%

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

      3. associate-*l* 59.7%

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

      4. unpow2 59.7%

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

      5. unpow2 59.7%

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

      6. difference-of-squares 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in angle around 0 58.5%

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

   5. Taylor expanded in a around 0 45.6%

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

      1. *-commutative 45.6%

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

   7. Simplified 45.6%

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

   if 800 < a < 5.80000000000000029e159

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

      3. associate-*l* 55.9%

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

      4. unpow2 55.9%

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

      5. unpow2 55.9%

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

      6. difference-of-squares 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in angle around 0 53.4%

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

   5. Taylor expanded in a around inf 38.4%

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

      1. *-commutative 38.4%

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

   7. Simplified 38.4%

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

   if 5.80000000000000029e159 < a

   1. Initial program 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*l* 51.5%

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

      3. associate-*l* 51.5%

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

      4. unpow2 51.5%

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

      5. unpow2 51.5%

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

      6. difference-of-squares 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in angle around 0 56.1%

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

   5. Taylor expanded in b around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. *-commutative 56.1%

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

      3. unpow2 56.1%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in angle around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. associate-*r* 56.1%

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

      3. *-commutative 56.1%

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

      4. unpow2 56.1%

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

      5. *-commutative 56.1%

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

      6. associate-*l* 56.1%

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

      7. *-commutative 56.1%

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

      8. unpow2 56.1%

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

      9. *-commutative 56.1%

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

      10. unpow2 56.1%

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

   10. Simplified 56.1%

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

   11. Taylor expanded in angle around 0 56.1%

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

       1. unpow2 56.1%

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

       2. associate-*r* 77.1%

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

   13. Simplified 77.1%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 800:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 13: 58.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.8e+159)
   (* 0.011111111111111112 (* angle (* (- b a) (* (+ b a) PI))))
   (* PI (* -0.011111111111111112 (* a (* a angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.8e+159) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * ((double) M_PI))));
	} else {
		tmp = ((double) M_PI) * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.8e+159) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * Math.PI)));
	} else {
		tmp = Math.PI * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 5.8e+159:
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * math.pi)))
	else:
		tmp = math.pi * (-0.011111111111111112 * (a * (a * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.8e+159)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(Float64(b + a) * pi))));
	else
		tmp = Float64(pi * Float64(-0.011111111111111112 * Float64(a * Float64(a * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.8e+159)
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * pi)));
	else
		tmp = pi * (-0.011111111111111112 * (a * (a * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 5.8e+159], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(-0.011111111111111112 * N[(a * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.80000000000000029e159

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 59.1%

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

      3. associate-*l* 59.1%

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

      4. unpow2 59.1%

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

      5. unpow2 59.1%

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

      6. difference-of-squares 60.8%

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

   3. Simplified 60.8%

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

   4. Taylor expanded in angle around 0 57.6%

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

   if 5.80000000000000029e159 < a

   1. Initial program 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*l* 51.5%

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

      3. associate-*l* 51.5%

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

      4. unpow2 51.5%

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

      5. unpow2 51.5%

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

      6. difference-of-squares 60.6%

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

   3. Simplified 60.6%

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

   4. Taylor expanded in angle around 0 56.1%

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

   5. Taylor expanded in b around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. *-commutative 56.1%

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

      3. unpow2 56.1%

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

   7. Simplified 56.1%

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

   8. Taylor expanded in angle around 0 56.1%

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

      1. *-commutative 56.1%

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

      2. associate-*r* 56.1%

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

      3. *-commutative 56.1%

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

      4. unpow2 56.1%

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

      5. *-commutative 56.1%

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

      6. associate-*l* 56.1%

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

      7. *-commutative 56.1%

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

      8. unpow2 56.1%

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

      9. *-commutative 56.1%

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

      10. unpow2 56.1%

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

   10. Simplified 56.1%

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

   11. Taylor expanded in angle around 0 56.1%

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

       1. unpow2 56.1%

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

       2. associate-*r* 77.1%

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

   13. Simplified 77.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 14: 59.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+153}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.005555555555555556 \cdot \left(\left(a \cdot angle\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.35e+153)
   (* 0.011111111111111112 (* angle (* (- b a) (* (+ b a) PI))))
   (* 2.0 (* 0.005555555555555556 (* (* a angle) (* (- b a) PI))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.35e+153) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * ((double) M_PI))));
	} else {
		tmp = 2.0 * (0.005555555555555556 * ((a * angle) * ((b - a) * ((double) M_PI))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.35e+153) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * Math.PI)));
	} else {
		tmp = 2.0 * (0.005555555555555556 * ((a * angle) * ((b - a) * Math.PI)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 1.35e+153:
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * math.pi)))
	else:
		tmp = 2.0 * (0.005555555555555556 * ((a * angle) * ((b - a) * math.pi)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.35e+153)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(Float64(b + a) * pi))));
	else
		tmp = Float64(2.0 * Float64(0.005555555555555556 * Float64(Float64(a * angle) * Float64(Float64(b - a) * pi))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.35e+153)
		tmp = 0.011111111111111112 * (angle * ((b - a) * ((b + a) * pi)));
	else
		tmp = 2.0 * (0.005555555555555556 * ((a * angle) * ((b - a) * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1.35e+153], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.005555555555555556 * N[(N[(a * angle), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.35e153

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-*l* 59.6%

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

      3. associate-*l* 59.6%

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

      4. unpow2 59.6%

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

      5. unpow2 59.6%

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

      6. difference-of-squares 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in angle around 0 57.6%

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

   if 1.35e153 < a

   1. Initial program 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-*l* 47.2%

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

      3. associate-*l* 47.2%

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

      4. unpow2 47.2%

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

      5. unpow2 47.2%

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

      6. difference-of-squares 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in angle around inf 79.0%

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

      1. associate-*r* 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*r* 80.2%

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

      4. *-commutative 80.2%

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

      5. *-commutative 80.2%

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

      6. *-commutative 80.2%

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

      7. associate-*r* 79.1%

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

      8. *-commutative 79.1%

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

      9. *-commutative 79.1%

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

      10. *-commutative 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in a around inf 75.6%

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

   8. Taylor expanded in angle around 0 52.0%

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

      1. associate-*r* 71.2%

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

      2. *-commutative 71.2%

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

   10. Simplified 71.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{+153}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.005555555555555556 \cdot \left(\left(a \cdot angle\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 15: 62.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.15 \cdot 10^{+43}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.15e+43)
   (* (+ b a) (* 0.011111111111111112 (* PI (* (- b a) angle))))
   (* 0.011111111111111112 (* angle (* (- b a) (* a PI))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.15e+43) {
		tmp = (b + a) * (0.011111111111111112 * (((double) M_PI) * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.15e+43) {
		tmp = (b + a) * (0.011111111111111112 * (Math.PI * ((b - a) * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * Math.PI)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 2.15e+43:
		tmp = (b + a) * (0.011111111111111112 * (math.pi * ((b - a) * angle)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * math.pi)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.15e+43)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(b - a) * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(a * pi))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.15e+43)
		tmp = (b + a) * (0.011111111111111112 * (pi * ((b - a) * angle)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[angle, 2.15e+43], 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[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.15e43

   1. Initial program 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*l* 65.6%

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

      3. associate-*l* 65.6%

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

      4. unpow2 65.6%

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

      5. unpow2 65.6%

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

      6. difference-of-squares 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in angle around inf 65.3%

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

      1. expm1-log1p-u 50.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      2. expm1-udef 32.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} - 1} \]

   6. Applied egg-rr 38.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 56.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\right)} \]

      2. expm1-log1p 78.5%

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

      3. *-commutative 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in angle around 0 76.5%

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

       1. associate-*r* 76.5%

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

       2. *-commutative 76.5%

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

   11. Simplified 76.5%

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

   if 2.15e43 < angle

   1. Initial program 35.2%

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

      1. *-commutative 35.2%

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

      2. associate-*l* 35.2%

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

      3. associate-*l* 35.2%

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

      4. unpow2 35.2%

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

      5. unpow2 35.2%

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

      6. difference-of-squares 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in angle around 0 32.0%

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

   5. Taylor expanded in a around inf 27.5%

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

      1. *-commutative 27.5%

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

   7. Simplified 27.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.15 \cdot 10^{+43}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b - a\right) \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 16: 40.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-29}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 6e-29)
   (* PI (* -0.011111111111111112 (* angle (* a a))))
   (* -0.011111111111111112 (* PI (* a (* a angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6e-29) {
		tmp = ((double) M_PI) * (-0.011111111111111112 * (angle * (a * a)));
	} else {
		tmp = -0.011111111111111112 * (((double) M_PI) * (a * (a * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6e-29) {
		tmp = Math.PI * (-0.011111111111111112 * (angle * (a * a)));
	} else {
		tmp = -0.011111111111111112 * (Math.PI * (a * (a * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 6e-29:
		tmp = math.pi * (-0.011111111111111112 * (angle * (a * a)))
	else:
		tmp = -0.011111111111111112 * (math.pi * (a * (a * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 6e-29)
		tmp = Float64(pi * Float64(-0.011111111111111112 * Float64(angle * Float64(a * a))));
	else
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(a * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 6e-29)
		tmp = pi * (-0.011111111111111112 * (angle * (a * a)));
	else
		tmp = -0.011111111111111112 * (pi * (a * (a * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 6e-29], N[(Pi * N[(-0.011111111111111112 * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.011111111111111112 * N[(Pi * N[(a * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.0000000000000005e-29

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 59.1%

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

      3. associate-*l* 59.1%

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

      4. unpow2 59.1%

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

      5. unpow2 59.1%

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

      6. difference-of-squares 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in angle around 0 57.8%

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

   5. Taylor expanded in b around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. *-commutative 37.5%

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

      3. unpow2 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in angle around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. associate-*r* 37.5%

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

      3. *-commutative 37.5%

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

      4. unpow2 37.5%

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

      5. *-commutative 37.5%

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

      6. associate-*l* 37.5%

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

      7. *-commutative 37.5%

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

      8. unpow2 37.5%

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

      9. *-commutative 37.5%

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

      10. unpow2 37.5%

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

   10. Simplified 37.5%

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

   if 6.0000000000000005e-29 < a

   1. Initial program 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*l* 56.9%

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

      3. associate-*l* 56.9%

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

      4. unpow2 56.9%

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

      5. unpow2 56.9%

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

      6. difference-of-squares 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in angle around 0 56.8%

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

   5. Taylor expanded in b around 0 35.9%

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

      1. *-commutative 35.9%

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

      2. *-commutative 35.9%

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

      3. unpow2 35.9%

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

   7. Simplified 35.9%

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

   8. Taylor expanded in angle around 0 35.9%

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

      1. *-commutative 35.9%

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

      2. associate-*r* 35.9%

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

      3. *-commutative 35.9%

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

      4. unpow2 35.9%

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

      5. *-commutative 35.9%

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

      6. associate-*l* 35.9%

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

      7. *-commutative 35.9%

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

      8. unpow2 35.9%

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

      9. *-commutative 35.9%

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

      10. unpow2 35.9%

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

   10. Simplified 35.9%

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

   11. Taylor expanded in angle around 0 35.9%

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

       1. associate-*r* 35.9%

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

       2. unpow2 35.9%

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

       3. associate-*r* 42.4%

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

   13. Simplified 42.4%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-29}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 17: 40.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+141}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e+141)
   (* (* angle -0.011111111111111112) (* PI (* a a)))
   (* PI (* -0.011111111111111112 (* a (* a angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.2e+141) {
		tmp = (angle * -0.011111111111111112) * (((double) M_PI) * (a * a));
	} else {
		tmp = ((double) M_PI) * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.2e+141) {
		tmp = (angle * -0.011111111111111112) * (Math.PI * (a * a));
	} else {
		tmp = Math.PI * (-0.011111111111111112 * (a * (a * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 1.2e+141:
		tmp = (angle * -0.011111111111111112) * (math.pi * (a * a))
	else:
		tmp = math.pi * (-0.011111111111111112 * (a * (a * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.2e+141)
		tmp = Float64(Float64(angle * -0.011111111111111112) * Float64(pi * Float64(a * a)));
	else
		tmp = Float64(pi * Float64(-0.011111111111111112 * Float64(a * Float64(a * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.2e+141)
		tmp = (angle * -0.011111111111111112) * (pi * (a * a));
	else
		tmp = pi * (-0.011111111111111112 * (a * (a * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1.2e+141], N[(N[(angle * -0.011111111111111112), $MachinePrecision] * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(-0.011111111111111112 * N[(a * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.19999999999999999e141

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. associate-*l* 59.6%

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

      3. associate-*l* 59.6%

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

      4. unpow2 59.6%

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

      5. unpow2 59.6%

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

      6. difference-of-squares 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in angle around 0 57.6%

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

   5. Taylor expanded in b around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. *-commutative 35.7%

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

      3. unpow2 35.7%

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

   7. Simplified 35.7%

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

   8. Taylor expanded in angle around 0 35.7%

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

      1. associate-*r* 35.7%

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

      2. *-commutative 35.7%

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

      3. unpow2 35.7%

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

   10. Simplified 35.7%

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

   if 1.19999999999999999e141 < a

   1. Initial program 50.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 50.0%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 50.0%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 50.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 50.0%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 50.0%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 59.7%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 59.7%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 57.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 47.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 47.3%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 47.3%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 47.3%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 47.3%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. associate-*r* 47.3%

         \[\leadsto \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \cdot -0.011111111111111112 \]

      3. *-commutative 47.3%

         \[\leadsto \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \pi\right) \cdot -0.011111111111111112 \]

      4. unpow2 47.3%

         \[\leadsto \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \pi\right) \cdot -0.011111111111111112 \]

      5. *-commutative 47.3%

         \[\leadsto \color{blue}{\left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \cdot -0.011111111111111112 \]

      6. associate-*l* 47.3%

         \[\leadsto \color{blue}{\pi \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot -0.011111111111111112\right)} \]

      7. *-commutative 47.3%

         \[\leadsto \pi \cdot \color{blue}{\left(-0.011111111111111112 \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \]

      8. unpow2 47.3%

         \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \left(\color{blue}{{a}^{2}} \cdot angle\right)\right) \]

      9. *-commutative 47.3%

         \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(angle \cdot {a}^{2}\right)}\right) \]

      10. unpow2 47.3%

          \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

   10. Simplified 47.3%

       \[\leadsto \color{blue}{\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)} \]

   11. Taylor expanded in angle around 0 47.3%

       \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(angle \cdot {a}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow2 47.3%

          \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       2. associate-*r* 62.2%

          \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot a\right) \cdot a\right)}\right) \]

   13. Simplified 62.2%

       \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot a\right) \cdot a\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{+141}:\\ \;\;\;\;\left(angle \cdot -0.011111111111111112\right) \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 18: 48.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 15000000000000:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 15000000000000.0)
   (* -0.011111111111111112 (* PI (* a (* a angle))))
   (* 0.011111111111111112 (* angle (* PI (* b b))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 15000000000000.0) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (a * (a * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 15000000000000.0) {
		tmp = -0.011111111111111112 * (Math.PI * (a * (a * angle)));
	} else {
		tmp = 0.011111111111111112 * (angle * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 15000000000000.0:
		tmp = -0.011111111111111112 * (math.pi * (a * (a * angle)))
	else:
		tmp = 0.011111111111111112 * (angle * (math.pi * (b * b)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 15000000000000.0)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(a * angle))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 15000000000000.0)
		tmp = -0.011111111111111112 * (pi * (a * (a * angle)));
	else
		tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 15000000000000.0], N[(-0.011111111111111112 * N[(Pi * N[(a * N[(a * 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 b < 1.5e13

   1. Initial program 59.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 59.5%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 59.5%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 59.5%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 59.5%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 59.5%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 61.7%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 60.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around 0 46.0%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. *-commutative 46.0%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      3. unpow2 46.0%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 46.0%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   8. Taylor expanded in angle around 0 46.0%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. associate-*r* 46.0%

         \[\leadsto \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \cdot -0.011111111111111112 \]

      3. *-commutative 46.0%

         \[\leadsto \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \pi\right) \cdot -0.011111111111111112 \]

      4. unpow2 46.0%

         \[\leadsto \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \pi\right) \cdot -0.011111111111111112 \]

      5. *-commutative 46.0%

         \[\leadsto \color{blue}{\left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \cdot -0.011111111111111112 \]

      6. associate-*l* 46.0%

         \[\leadsto \color{blue}{\pi \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot -0.011111111111111112\right)} \]

      7. *-commutative 46.0%

         \[\leadsto \pi \cdot \color{blue}{\left(-0.011111111111111112 \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \]

      8. unpow2 46.0%

         \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \left(\color{blue}{{a}^{2}} \cdot angle\right)\right) \]

      9. *-commutative 46.0%

         \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(angle \cdot {a}^{2}\right)}\right) \]

      10. unpow2 46.0%

          \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

   10. Simplified 46.0%

       \[\leadsto \color{blue}{\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)} \]

   11. Taylor expanded in angle around 0 46.0%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 46.0%

          \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \]

       2. unpow2 46.0%

          \[\leadsto -0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \pi\right) \]

       3. associate-*r* 47.9%

          \[\leadsto -0.011111111111111112 \cdot \left(\color{blue}{\left(\left(angle \cdot a\right) \cdot a\right)} \cdot \pi\right) \]

   13. Simplified 47.9%

       \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(angle \cdot a\right) \cdot a\right) \cdot \pi\right)} \]

   if 1.5e13 < b

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 55.5%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 55.5%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 55.5%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 55.5%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 55.5%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 58.4%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 50.3%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

   5. Taylor expanded in b around inf 46.1%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 46.1%

         \[\leadsto \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

      2. *-commutative 46.1%

         \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \cdot 0.011111111111111112 \]

      3. unpow2 46.1%

         \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \cdot 0.011111111111111112 \]

   7. Simplified 46.1%

      \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \cdot 0.011111111111111112} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15000000000000:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 19: 38.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ -0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* -0.011111111111111112 (* PI (* a (* a angle)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return -0.011111111111111112 * (((double) M_PI) * (a * (a * angle)));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return -0.011111111111111112 * (Math.PI * (a * (a * angle)));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return -0.011111111111111112 * (math.pi * (a * (a * angle)))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(-0.011111111111111112 * Float64(pi * Float64(a * Float64(a * angle))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = -0.011111111111111112 * (pi * (a * (a * angle)));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(-0.011111111111111112 * N[(Pi * N[(a * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. *-commutative 58.5%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 58.5%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 58.5%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 58.5%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 58.5%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 60.8%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 60.8%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 57.5%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)} \]

5. Taylor expanded in b around 0 37.1%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 37.1%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. *-commutative 37.1%

      \[\leadsto \left(angle \cdot \color{blue}{\left(\pi \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

   3. unpow2 37.1%

      \[\leadsto \left(angle \cdot \left(\pi \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

7. Simplified 37.1%

   \[\leadsto \color{blue}{\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

8. Taylor expanded in angle around 0 37.1%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 37.1%

      \[\leadsto \color{blue}{\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

   2. associate-*r* 37.1%

      \[\leadsto \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \cdot -0.011111111111111112 \]

   3. *-commutative 37.1%

      \[\leadsto \left(\color{blue}{\left({a}^{2} \cdot angle\right)} \cdot \pi\right) \cdot -0.011111111111111112 \]

   4. unpow2 37.1%

      \[\leadsto \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot angle\right) \cdot \pi\right) \cdot -0.011111111111111112 \]

   5. *-commutative 37.1%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \cdot -0.011111111111111112 \]

   6. associate-*l* 37.1%

      \[\leadsto \color{blue}{\pi \cdot \left(\left(\left(a \cdot a\right) \cdot angle\right) \cdot -0.011111111111111112\right)} \]

   7. *-commutative 37.1%

      \[\leadsto \pi \cdot \color{blue}{\left(-0.011111111111111112 \cdot \left(\left(a \cdot a\right) \cdot angle\right)\right)} \]

   8. unpow2 37.1%

      \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \left(\color{blue}{{a}^{2}} \cdot angle\right)\right) \]

   9. *-commutative 37.1%

      \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(angle \cdot {a}^{2}\right)}\right) \]

   10. unpow2 37.1%

       \[\leadsto \pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

10. Simplified 37.1%

    \[\leadsto \color{blue}{\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)} \]

11. Taylor expanded in angle around 0 37.1%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 37.1%

       \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {a}^{2}\right) \cdot \pi\right)} \]

    2. unpow2 37.1%

       \[\leadsto -0.011111111111111112 \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \pi\right) \]

    3. associate-*r* 39.2%

       \[\leadsto -0.011111111111111112 \cdot \left(\color{blue}{\left(\left(angle \cdot a\right) \cdot a\right)} \cdot \pi\right) \]

13. Simplified 39.2%

    \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(angle \cdot a\right) \cdot a\right) \cdot \pi\right)} \]

14. Final simplification 39.2%

    \[\leadsto -0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right) \]

Reproduce

herbie shell --seed 2023264 
(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)))))