ab-angle->ABCF B

Percentage Accurate: 54.6% → 67.7%

Time: 35.8s

Alternatives: 17

Speedup: 5.5×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+178}:\\ \;\;\;\;\left|2 \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)\right| \cdot \cos t_1\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+118}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos t_0 \cdot \left(\sin t_0 \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_1 \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{\frac{180}{angle}}\right)\right)\\ \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (/ angle 180.0) PI)))
   (if (<= (/ angle 180.0) -2e+178)
     (*
      (fabs
       (*
        2.0
        (* (sin (* angle (* 0.005555555555555556 PI))) (pow (hypot b a) 2.0))))
      (cos t_1))
     (if (<= (/ angle 180.0) 2e+118)
       (* 2.0 (* (- b a) (* (cos t_0) (* (sin t_0) (+ b a)))))
       (*
        (sin t_1)
        (*
         (* 2.0 (+ (* b b) (* a a)))
         (cos (* (pow (cbrt PI) 2.0) (/ (cbrt PI) (/ 180.0 angle))))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((angle / 180.0) <= -2e+178) {
		tmp = fabs((2.0 * (sin((angle * (0.005555555555555556 * ((double) M_PI)))) * pow(hypot(b, a), 2.0)))) * cos(t_1);
	} else if ((angle / 180.0) <= 2e+118) {
		tmp = 2.0 * ((b - a) * (cos(t_0) * (sin(t_0) * (b + a))));
	} else {
		tmp = sin(t_1) * ((2.0 * ((b * b) + (a * a))) * cos((pow(cbrt(((double) M_PI)), 2.0) * (cbrt(((double) M_PI)) / (180.0 / angle)))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = (angle / 180.0) * Math.PI;
	double tmp;
	if ((angle / 180.0) <= -2e+178) {
		tmp = Math.abs((2.0 * (Math.sin((angle * (0.005555555555555556 * Math.PI))) * Math.pow(Math.hypot(b, a), 2.0)))) * Math.cos(t_1);
	} else if ((angle / 180.0) <= 2e+118) {
		tmp = 2.0 * ((b - a) * (Math.cos(t_0) * (Math.sin(t_0) * (b + a))));
	} else {
		tmp = Math.sin(t_1) * ((2.0 * ((b * b) + (a * a))) * Math.cos((Math.pow(Math.cbrt(Math.PI), 2.0) * (Math.cbrt(Math.PI) / (180.0 / angle)))));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e+178)
		tmp = Float64(abs(Float64(2.0 * Float64(sin(Float64(angle * Float64(0.005555555555555556 * pi))) * (hypot(b, a) ^ 2.0)))) * cos(t_1));
	elseif (Float64(angle / 180.0) <= 2e+118)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(cos(t_0) * Float64(sin(t_0) * Float64(b + a)))));
	else
		tmp = Float64(sin(t_1) * Float64(Float64(2.0 * Float64(Float64(b * b) + Float64(a * a))) * cos(Float64((cbrt(pi) ^ 2.0) * Float64(cbrt(pi) / Float64(180.0 / angle))))));
	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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+178], N[(N[Abs[N[(2.0 * N[(N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e+118], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[(2.0 * N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e178

   1. Initial program 30.1%

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

      1. *-commutative 30.1%

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

      2. associate-*l* 30.1%

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

      3. unpow2 30.1%

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

      4. fma-neg 34.6%

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

      5. unpow2 34.6%

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

      6. distribute-rgt-neg-in 34.6%

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

   3. Simplified 34.6%

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

      1. clear-num 39.1%

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

      2. un-div-inv 35.4%

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

   5. Applied egg-rr 35.4%

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

      1. add-sqr-sqrt 31.9%

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

      2. sqrt-unprod 61.3%

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

      3. pow2 61.3%

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

   7. Applied egg-rr 61.1%

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

      1. unpow2 61.1%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right) \cdot \left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. rem-sqrt-square 61.1%

         \[\leadsto \color{blue}{\left|{\left(\mathsf{hypot}\left(b, a\right)\right)}^{2} \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right|} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. *-commutative 61.1%

         \[\leadsto \left|\color{blue}{\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}}\right| \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. *-commutative 61.1%

         \[\leadsto \left|\color{blue}{\left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right| \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. associate-*l* 61.1%

         \[\leadsto \left|\color{blue}{2 \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}\right| \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. *-commutative 61.1%

         \[\leadsto \left|2 \cdot \left(\sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)\right| \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. associate-*l* 61.1%

         \[\leadsto \left|2 \cdot \left(\sin \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)} \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)\right| \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   9. Simplified 61.1%

      \[\leadsto \color{blue}{\left|2 \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)\right|} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   if -2.0000000000000001e178 < (/.f64 angle 180) < 1.99999999999999993e118

   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. associate-*l* 65.4%

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

      2. unpow2 65.4%

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

      3. unpow2 65.4%

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

      4. difference-of-squares 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in angle around inf 83.6%

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

   if 1.99999999999999993e118 < (/.f64 angle 180)

   1. Initial program 31.2%

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

      1. *-commutative 31.2%

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

      2. associate-*l* 31.2%

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

      3. unpow2 31.2%

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

      4. fma-neg 34.1%

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

      5. unpow2 34.1%

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

   3. Simplified 34.1%

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

      1. fma-udef 31.2%

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

      2. add-sqr-sqrt 13.5%

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

      3. sqrt-unprod 39.9%

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

      4. sqr-neg 39.9%

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

      5. sqrt-unprod 40.1%

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

      6. add-sqr-sqrt 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. add-cbrt-cube 0.0%

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

      2. pow1/3 0.0%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \color{blue}{\left({\left(\left(\left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}^{0.3333333333333333}\right)}\right) \]

      3. pow3 0.0%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \left({\color{blue}{\left({\left(\pi \cdot \frac{angle}{180}\right)}^{3}\right)}}^{0.3333333333333333}\right)\right) \]

      4. div-inv 0.0%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \left({\left({\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \]

      5. metadata-eval 0.0%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \left({\left({\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right) \]

   7. Applied egg-rr 0.0%

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \color{blue}{\left({\left({\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow1/3 0.0%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\sqrt[3]{{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{3}}\right)}\right) \]

      2. rem-cbrt-cube 46.3%

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

      3. metadata-eval 46.3%

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

      4. div-inv 40.1%

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

      5. clear-num 45.5%

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

      6. div-inv 39.7%

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

      7. add-cube-cbrt 44.9%

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

      8. *-un-lft-identity 44.9%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \left(\frac{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}{\color{blue}{1 \cdot \frac{180}{angle}}}\right)\right) \]

      9. times-frac 49.0%

         \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{1} \cdot \frac{\sqrt[3]{\pi}}{\frac{180}{angle}}\right)}\right) \]

      10. pow2 49.0%

          \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \left(\frac{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\pi}}{\frac{180}{angle}}\right)\right) \]

   9. Applied egg-rr 49.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{+178}:\\ \;\;\;\;\left|2 \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot {\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)\right| \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+118}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{\frac{180}{angle}}\right)\right)\\ \end{array} \]

Alternative 2: 68.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \sin t_0\\ t_2 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\cos t_2 \cdot \left(\sin t_2 \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \leq -2 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(t_1 \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(t_1 \cdot \left(\left(b + a\right) \cdot \cos t_0\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* PI (* angle 0.005555555555555556)))
        (t_1 (sin t_0))
        (t_2 (* (/ angle 180.0) PI)))
   (if (<=
        (* (cos t_2) (* (sin t_2) (* 2.0 (- (pow b 2.0) (pow a 2.0)))))
        -2e+154)
     (*
      2.0
      (*
       (* (- b a) (cos (* (pow (sqrt PI) 2.0) (* angle 0.005555555555555556))))
       (* t_1 (+ b a))))
     (* 2.0 (* (- b a) (* t_1 (* (+ b a) (cos t_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 t_1 = sin(t_0);
	double t_2 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((cos(t_2) * (sin(t_2) * (2.0 * (pow(b, 2.0) - pow(a, 2.0))))) <= -2e+154) {
		tmp = 2.0 * (((b - a) * cos((pow(sqrt(((double) M_PI)), 2.0) * (angle * 0.005555555555555556)))) * (t_1 * (b + a)));
	} else {
		tmp = 2.0 * ((b - a) * (t_1 * ((b + a) * cos(t_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 t_1 = Math.sin(t_0);
	double t_2 = (angle / 180.0) * Math.PI;
	double tmp;
	if ((Math.cos(t_2) * (Math.sin(t_2) * (2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))))) <= -2e+154) {
		tmp = 2.0 * (((b - a) * Math.cos((Math.pow(Math.sqrt(Math.PI), 2.0) * (angle * 0.005555555555555556)))) * (t_1 * (b + a)));
	} else {
		tmp = 2.0 * ((b - a) * (t_1 * ((b + a) * Math.cos(t_0))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (angle * 0.005555555555555556)
	t_1 = math.sin(t_0)
	t_2 = (angle / 180.0) * math.pi
	tmp = 0
	if (math.cos(t_2) * (math.sin(t_2) * (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))))) <= -2e+154:
		tmp = 2.0 * (((b - a) * math.cos((math.pow(math.sqrt(math.pi), 2.0) * (angle * 0.005555555555555556)))) * (t_1 * (b + a)))
	else:
		tmp = 2.0 * ((b - a) * (t_1 * ((b + a) * math.cos(t_0))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_1 = sin(t_0)
	t_2 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64(cos(t_2) * Float64(sin(t_2) * Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))))) <= -2e+154)
		tmp = Float64(2.0 * Float64(Float64(Float64(b - a) * cos(Float64((sqrt(pi) ^ 2.0) * Float64(angle * 0.005555555555555556)))) * Float64(t_1 * Float64(b + a))));
	else
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(t_1 * Float64(Float64(b + a) * cos(t_0)))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle * 0.005555555555555556);
	t_1 = sin(t_0);
	t_2 = (angle / 180.0) * pi;
	tmp = 0.0;
	if ((cos(t_2) * (sin(t_2) * (2.0 * ((b ^ 2.0) - (a ^ 2.0))))) <= -2e+154)
		tmp = 2.0 * (((b - a) * cos(((sqrt(pi) ^ 2.0) * (angle * 0.005555555555555556)))) * (t_1 * (b + a)));
	else
		tmp = 2.0 * ((b - a) * (t_1 * ((b + a) * cos(t_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[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(N[Cos[t$95$2], $MachinePrecision] * N[(N[Sin[t$95$2], $MachinePrecision] * N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+154], N[(2.0 * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(t$95$1 * N[(N[(b + a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) < -2.00000000000000007e154

   1. Initial program 40.6%

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

      1. associate-*l* 40.6%

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

      2. unpow2 40.6%

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

      3. unpow2 40.6%

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

      4. difference-of-squares 40.6%

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

   3. Simplified 40.6%

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

   4. Taylor expanded in angle around inf 64.6%

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

      1. associate-*r* 64.6%

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

      2. *-commutative 64.6%

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

      3. *-commutative 64.6%

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

      4. associate-*r* 63.3%

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

      5. *-commutative 63.3%

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

      6. *-commutative 63.3%

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

      7. *-commutative 63.3%

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

      8. associate-*r* 61.3%

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

      9. *-commutative 61.3%

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

      10. +-commutative 61.3%

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

   6. Simplified 61.3%

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

      1. add-sqr-sqrt 68.0%

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

      2. pow2 68.0%

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

   8. Applied egg-rr 68.0%

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

   if -2.00000000000000007e154 < (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))

   1. Initial program 63.9%

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

      1. associate-*l* 63.9%

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

      2. unpow2 63.9%

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

      3. unpow2 63.9%

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

      4. difference-of-squares 70.3%

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

   3. Simplified 70.3%

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

      1. add-cube-cbrt 69.7%

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

      2. pow3 69.8%

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

      3. div-inv 69.9%

         \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left({\left(\sqrt[3]{\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}\right)}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. metadata-eval 69.9%

         \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left({\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}\right)}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. Applied egg-rr 69.9%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. Taylor expanded in angle around inf 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. associate-*r* 74.8%

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

      4. *-commutative 74.8%

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

      5. associate-*r* 77.1%

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

      6. *-commutative 77.1%

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

      7. associate-*l* 77.1%

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

      8. *-commutative 77.1%

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

      9. *-commutative 77.1%

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

      10. +-commutative 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \leq -2 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\left(\left(b - a\right) \cdot \cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b + a\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= (pow a 2.0) 2e+297)
   (*
    2.0
    (*
     (* (sin (* PI (* angle 0.005555555555555556))) (+ b a))
     (* (- b a) (cos (* 0.005555555555555556 (* angle PI))))))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (pow(a, 2.0) <= 2e+297) {
		tmp = 2.0 * ((sin((((double) M_PI) * (angle * 0.005555555555555556))) * (b + a)) * ((b - a) * cos((0.005555555555555556 * (angle * ((double) M_PI))))));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(a, 2.0) <= 2e+297) {
		tmp = 2.0 * ((Math.sin((Math.PI * (angle * 0.005555555555555556))) * (b + a)) * ((b - a) * Math.cos((0.005555555555555556 * (angle * Math.PI)))));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if math.pow(a, 2.0) <= 2e+297:
		tmp = 2.0 * ((math.sin((math.pi * (angle * 0.005555555555555556))) * (b + a)) * ((b - a) * math.cos((0.005555555555555556 * (angle * math.pi)))))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if ((a ^ 2.0) <= 2e+297)
		tmp = Float64(2.0 * Float64(Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(0.005555555555555556 * Float64(angle * pi))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a ^ 2.0) <= 2e+297)
		tmp = 2.0 * ((sin((pi * (angle * 0.005555555555555556))) * (b + a)) * ((b - a) * cos((0.005555555555555556 * (angle * pi)))));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	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[Power[a, 2.0], $MachinePrecision], 2e+297], N[(2.0 * N[(N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 2e297

   1. Initial program 69.8%

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

      1. associate-*l* 69.8%

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

      2. unpow2 69.8%

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

      3. unpow2 69.8%

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

      4. difference-of-squares 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in angle around inf 71.9%

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

      1. associate-*r* 71.9%

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

      2. *-commutative 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-*r* 72.2%

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

      5. *-commutative 72.2%

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

      6. *-commutative 72.2%

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

      7. *-commutative 72.2%

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

      8. associate-*r* 71.5%

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

      9. *-commutative 71.5%

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

      10. +-commutative 71.5%

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

   6. Simplified 71.5%

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

   7. Taylor expanded in angle around inf 70.9%

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

   if 2e297 < (pow.f64 a 2)

   1. Initial program 29.3%

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

      1. associate-*l* 29.3%

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

      2. unpow2 29.3%

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

      3. unpow2 29.3%

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

      4. difference-of-squares 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in angle around 0 50.8%

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

      1. associate-*r* 77.9%

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

      2. *-commutative 77.9%

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

      3. +-commutative 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;2 \cdot \left(\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 4: 68.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ 2 \cdot \left(\left(\sin t_0 \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos t_0\right)\right) \end{array} \end{array} \]

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))))
   (* 2.0 (* (* (sin t_0) (+ b a)) (* (- b a) (cos t_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);
	return 2.0 * ((sin(t_0) * (b + a)) * ((b - a) * cos(t_0)));
}

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);
	return 2.0 * ((Math.sin(t_0) * (b + a)) * ((b - a) * Math.cos(t_0)));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (angle * 0.005555555555555556)
	return 2.0 * ((math.sin(t_0) * (b + a)) * ((b - a) * math.cos(t_0)))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	return Float64(2.0 * Float64(Float64(sin(t_0) * Float64(b + a)) * Float64(Float64(b - a) * cos(t_0))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	t_0 = pi * (angle * 0.005555555555555556);
	tmp = 2.0 * ((sin(t_0) * (b + a)) * ((b - a) * cos(t_0)));
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]}, N[(2.0 * N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. unpow2 57.8%

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

   3. unpow2 57.8%

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

   4. difference-of-squares 62.5%

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

3. Simplified 62.5%

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

4. Taylor expanded in angle around inf 71.6%

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

   1. associate-*r* 71.6%

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

   2. *-commutative 71.6%

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

   3. *-commutative 71.6%

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

   4. associate-*r* 73.4%

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

   5. *-commutative 73.4%

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

   6. *-commutative 73.4%

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

   7. *-commutative 73.4%

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

   8. associate-*r* 73.0%

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

   9. *-commutative 73.0%

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

   10. +-commutative 73.0%

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

6. Simplified 73.0%

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

7. Final simplification 73.0%

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

Alternative 5: 68.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ 2 \cdot \left(\left(b - a\right) \cdot \left(\sin t_0 \cdot \left(\left(b + a\right) \cdot \cos t_0\right)\right)\right) \end{array} \end{array} \]

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
 (let* ((t_0 (* PI (* angle 0.005555555555555556))))
   (* 2.0 (* (- b a) (* (sin t_0) (* (+ b a) (cos t_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);
	return 2.0 * ((b - a) * (sin(t_0) * ((b + a) * cos(t_0))));
}

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);
	return 2.0 * ((b - a) * (Math.sin(t_0) * ((b + a) * Math.cos(t_0))));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.pi * (angle * 0.005555555555555556)
	return 2.0 * ((b - a) * (math.sin(t_0) * ((b + a) * math.cos(t_0))))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle * 0.005555555555555556))
	return Float64(2.0 * Float64(Float64(b - a) * Float64(sin(t_0) * Float64(Float64(b + a) * cos(t_0)))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	t_0 = pi * (angle * 0.005555555555555556);
	tmp = 2.0 * ((b - a) * (sin(t_0) * ((b + a) * cos(t_0))));
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]}, N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. unpow2 57.8%

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

   3. unpow2 57.8%

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

   4. difference-of-squares 62.5%

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

3. Simplified 62.5%

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

   1. add-cube-cbrt 62.1%

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

   2. pow3 62.1%

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

   3. div-inv 62.2%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left({\left(\sqrt[3]{\sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}\right)}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   4. metadata-eval 62.2%

      \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left({\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}\right)}^{3} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

5. Applied egg-rr 62.2%

   \[\leadsto \left(2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

6. Taylor expanded in angle around inf 71.6%

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

   1. associate-*r* 71.6%

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

   2. *-commutative 71.6%

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

   3. associate-*r* 71.0%

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

   4. *-commutative 71.0%

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

   5. associate-*r* 72.9%

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

   6. *-commutative 72.9%

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

   7. associate-*l* 73.0%

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

   8. *-commutative 73.0%

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

   9. *-commutative 73.0%

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

   10. +-commutative 73.0%

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

8. Simplified 73.0%

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

9. Final simplification 73.0%

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

Alternative 6: 64.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-223}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= (pow a 2.0) 2e-223)
   (* 2.0 (* (- b a) (* (sin (* PI (* angle 0.005555555555555556))) (+ b a))))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (pow(a, 2.0) <= 2e-223) {
		tmp = 2.0 * ((b - a) * (sin((((double) M_PI) * (angle * 0.005555555555555556))) * (b + a)));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.pow(a, 2.0) <= 2e-223) {
		tmp = 2.0 * ((b - a) * (Math.sin((Math.PI * (angle * 0.005555555555555556))) * (b + a)));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if math.pow(a, 2.0) <= 2e-223:
		tmp = 2.0 * ((b - a) * (math.sin((math.pi * (angle * 0.005555555555555556))) * (b + a)))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if ((a ^ 2.0) <= 2e-223)
		tmp = Float64(2.0 * Float64(Float64(b - a) * Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * Float64(b + a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((a ^ 2.0) <= 2e-223)
		tmp = 2.0 * ((b - a) * (sin((pi * (angle * 0.005555555555555556))) * (b + a)));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	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[Power[a, 2.0], $MachinePrecision], 2e-223], N[(2.0 * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 a 2) < 1.9999999999999999e-223

   1. Initial program 74.8%

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

      1. associate-*l* 74.8%

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

      2. unpow2 74.8%

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

      3. unpow2 74.8%

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

      4. difference-of-squares 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in angle around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. *-commutative 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-*r* 77.1%

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

      5. *-commutative 77.1%

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

      6. *-commutative 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-*r* 78.0%

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

      9. *-commutative 78.0%

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

      10. +-commutative 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in angle around 0 74.6%

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

   if 1.9999999999999999e-223 < (pow.f64 a 2)

   1. Initial program 51.1%

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

      1. associate-*l* 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

      4. difference-of-squares 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in angle around 0 57.9%

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

      1. associate-*r* 70.1%

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

      2. *-commutative 70.1%

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

      3. +-commutative 70.1%

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

   6. Simplified 70.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{a}^{2} \leq 2 \cdot 10^{-223}:\\ \;\;\;\;2 \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 7: 62.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \end{array} \]

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 4.2e-93)
   (* 2.0 (* (sin (* angle (* 0.005555555555555556 PI))) (* b b)))
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.2e-93) {
		tmp = 2.0 * (sin((angle * (0.005555555555555556 * ((double) M_PI)))) * (b * b));
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	}
	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 <= 4.2e-93) {
		tmp = 2.0 * (Math.sin((angle * (0.005555555555555556 * Math.PI))) * (b * b));
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 4.2e-93:
		tmp = 2.0 * (math.sin((angle * (0.005555555555555556 * math.pi))) * (b * b))
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.2e-93)
		tmp = Float64(2.0 * Float64(sin(Float64(angle * Float64(0.005555555555555556 * pi))) * Float64(b * b)));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.2e-93)
		tmp = 2.0 * (sin((angle * (0.005555555555555556 * pi))) * (b * b));
	else
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	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, 4.2e-93], N[(2.0 * N[(N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.2000000000000002e-93

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-*l* 61.5%

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

      3. unpow2 61.5%

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

      4. fma-neg 63.4%

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

      5. unpow2 63.4%

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

      6. distribute-rgt-neg-in 63.4%

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

   3. Simplified 63.4%

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

      1. clear-num 62.6%

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

      2. un-div-inv 61.7%

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

   5. Applied egg-rr 61.7%

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

   6. Taylor expanded in angle around 0 57.0%

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

   7. Taylor expanded in b around inf 43.2%

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

      1. unpow2 43.2%

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

      2. *-commutative 43.2%

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

      3. *-commutative 43.2%

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

      4. associate-*r* 45.5%

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

      5. *-commutative 45.5%

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

   9. Simplified 45.5%

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

   if 4.2000000000000002e-93 < a

   1. Initial program 50.8%

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

      1. associate-*l* 50.8%

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

      2. unpow2 50.8%

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

      3. unpow2 50.8%

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

      4. difference-of-squares 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in angle around 0 60.8%

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

      1. associate-*r* 71.5%

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

      2. *-commutative 71.5%

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

      3. +-commutative 71.5%

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

   6. Simplified 71.5%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \end{array} \]

Alternative 8: 62.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 7.8 \cdot 10^{+148}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left|\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right|\\ \end{array} \end{array} \]

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 7.8e+148)
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))
   (* angle (fabs (* PI (* (* a a) -0.011111111111111112))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 7.8e+148) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	} else {
		tmp = angle * fabs((((double) M_PI) * ((a * a) * -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 (angle <= 7.8e+148) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	} else {
		tmp = angle * Math.abs((Math.PI * ((a * a) * -0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 7.8e+148:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	else:
		tmp = angle * math.fabs((math.pi * ((a * a) * -0.011111111111111112)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 7.8e+148)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	else
		tmp = Float64(angle * abs(Float64(pi * Float64(Float64(a * a) * -0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 7.8e+148)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	else
		tmp = angle * abs((pi * ((a * a) * -0.011111111111111112)));
	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, 7.8e+148], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[Abs[N[(Pi * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.80000000000000004e148

   1. Initial program 61.2%

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

      1. associate-*l* 61.2%

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

      2. unpow2 61.2%

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

      3. unpow2 61.2%

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

      4. difference-of-squares 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in angle around 0 63.5%

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

      1. associate-*r* 73.7%

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

      2. *-commutative 73.7%

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

      3. +-commutative 73.7%

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

   6. Simplified 73.7%

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

   if 7.80000000000000004e148 < angle

   1. Initial program 27.6%

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

      1. associate-*l* 27.6%

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

      2. unpow2 27.6%

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

      3. unpow2 27.6%

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

      4. difference-of-squares 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in angle around 0 28.6%

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

   5. Taylor expanded in b around 0 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. *-commutative 25.0%

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

      4. unpow2 25.0%

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

   7. Simplified 25.0%

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

   8. Taylor expanded in angle around 0 25.0%

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

      1. *-commutative 25.0%

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

      2. unpow2 25.0%

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

      3. *-commutative 25.0%

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

      4. associate-*r* 25.0%

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

      5. associate-*r* 25.0%

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

      6. associate-*l* 25.0%

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

      7. *-commutative 25.0%

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

   10. Simplified 25.0%

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

       1. add-sqr-sqrt 11.6%

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

       2. sqrt-unprod 39.8%

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

       3. pow2 39.8%

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

       4. *-commutative 39.8%

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

   12. Applied egg-rr 39.8%

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

       1. unpow2 39.8%

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

       2. rem-sqrt-square 39.7%

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

       3. associate-*r* 39.7%

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

       4. *-commutative 39.7%

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

   14. Simplified 39.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 7.8 \cdot 10^{+148}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left|\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right|\\ \end{array} \]

Alternative 9: 63.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.02 \cdot 10^{+135}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \]

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 1.02e+135)
   (* 0.011111111111111112 (* (* PI (+ b a)) (* angle (- b a))))
   (* (* PI (+ (* b b) (* a a))) (* angle 0.011111111111111112))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 1.02e+135) {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b + a)) * (angle * (b - a)));
	} else {
		tmp = (((double) M_PI) * ((b * b) + (a * a))) * (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 (angle <= 1.02e+135) {
		tmp = 0.011111111111111112 * ((Math.PI * (b + a)) * (angle * (b - a)));
	} else {
		tmp = (Math.PI * ((b * b) + (a * a))) * (angle * 0.011111111111111112);
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 1.02e+135:
		tmp = 0.011111111111111112 * ((math.pi * (b + a)) * (angle * (b - a)))
	else:
		tmp = (math.pi * ((b * b) + (a * a))) * (angle * 0.011111111111111112)
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 1.02e+135)
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b + a)) * Float64(angle * Float64(b - a))));
	else
		tmp = Float64(Float64(pi * Float64(Float64(b * b) + Float64(a * a))) * Float64(angle * 0.011111111111111112));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 1.02e+135)
		tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
	else
		tmp = (pi * ((b * b) + (a * a))) * (angle * 0.011111111111111112);
	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, 1.02e+135], N[(0.011111111111111112 * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.01999999999999993e135

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. unpow2 61.1%

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

      3. unpow2 61.1%

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

      4. difference-of-squares 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in angle around 0 63.0%

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

      1. associate-*r* 73.3%

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

      2. *-commutative 73.3%

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

      3. +-commutative 73.3%

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

   6. Simplified 73.3%

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

   if 1.01999999999999993e135 < angle

   1. Initial program 31.7%

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

      1. *-commutative 31.7%

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

      2. associate-*l* 31.7%

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

      3. unpow2 31.7%

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

      4. fma-neg 35.1%

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

      5. unpow2 35.1%

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

   3. Simplified 35.1%

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

      1. fma-udef 31.7%

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

      2. add-sqr-sqrt 14.3%

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

      3. sqrt-unprod 41.9%

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

      4. sqr-neg 41.9%

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

      5. sqrt-unprod 42.1%

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

      6. add-sqr-sqrt 42.1%

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

   5. Applied egg-rr 42.1%

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

   6. Taylor expanded in angle around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. *-commutative 45.8%

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

      3. +-commutative 45.8%

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

      4. unpow2 45.8%

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

      5. unpow2 45.8%

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

   8. Simplified 45.8%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.02 \cdot 10^{+135}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 10: 50.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+18}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

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 2e+18)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (* 0.011111111111111112 (* angle (* (- b a) (* PI a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2e+18) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * a)));
	}
	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 <= 2e+18) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * a)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 2e+18:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * a)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 2e+18)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * a))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2e+18)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * a)));
	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, 2e+18], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2e18

   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. associate-*l* 62.5%

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

      2. unpow2 62.5%

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

      3. unpow2 62.5%

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

      4. difference-of-squares 65.2%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in angle around 0 60.5%

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

   5. Taylor expanded in b around inf 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-*r* 45.9%

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

      3. unpow2 45.9%

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

   7. Simplified 45.9%

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

   if 2e18 < a

   1. Initial program 43.9%

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

      1. associate-*l* 43.9%

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

      2. unpow2 43.9%

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

      3. unpow2 43.9%

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

      4. difference-of-squares 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in angle around 0 58.2%

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

   5. Taylor expanded in a around inf 47.8%

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

      1. *-commutative 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+18}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 11: 50.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

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 5.6e-30)
   (* 0.011111111111111112 (* angle (* (- b a) (* PI a))))
   (* 0.011111111111111112 (* angle (* (- b a) (* PI b))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.6e-30) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (((double) M_PI) * 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 <= 5.6e-30) {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * a)));
	} else {
		tmp = 0.011111111111111112 * (angle * ((b - a) * (Math.PI * b)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 5.6e-30:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * a)))
	else:
		tmp = 0.011111111111111112 * (angle * ((b - a) * (math.pi * b)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.6e-30)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * Float64(pi * b))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5.6e-30)
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * a)));
	else
		tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * 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, 5.6e-30], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.59999999999999977e-30

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. unpow2 58.2%

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

      3. unpow2 58.2%

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

      4. difference-of-squares 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in angle around 0 61.3%

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

   5. Taylor expanded in a around inf 46.5%

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

      1. *-commutative 46.5%

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

   7. Simplified 46.5%

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

   if 5.59999999999999977e-30 < b

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. unpow2 56.6%

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

      3. unpow2 56.6%

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

      4. difference-of-squares 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in angle around 0 55.8%

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

   5. Taylor expanded in a around 0 54.3%

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

      1. *-commutative 54.3%

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

   7. Simplified 54.3%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 12: 54.6% accurate, 5.5× speedup

Math

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

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 (* angle (* (- b a) (* PI (+ b a))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * ((b - a) * (pi * (b + a))));
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[(angle * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. unpow2 57.8%

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

   3. unpow2 57.8%

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

   4. difference-of-squares 62.5%

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

3. Simplified 62.5%

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

4. Taylor expanded in angle around 0 59.9%

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

5. Final simplification 59.9%

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

Alternative 13: 61.9% accurate, 5.5× speedup

Math

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

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 (+ b a)) (* angle (- b a)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((pi * (b + a)) * (angle * (b - a)));
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[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. unpow2 57.8%

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

   3. unpow2 57.8%

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

   4. difference-of-squares 62.5%

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

3. Simplified 62.5%

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

4. Taylor expanded in angle around 0 59.9%

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

   1. associate-*r* 68.3%

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

   2. *-commutative 68.3%

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

   3. +-commutative 68.3%

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

6. Simplified 68.3%

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

7. Final simplification 68.3%

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

Alternative 14: 47.4% accurate, 5.6× speedup

Math

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

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 (<= a 5e+49)
   (* (* PI (* b b)) (* angle 0.011111111111111112))
   (* angle (* PI (* a (* a -0.011111111111111112))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5e+49) {
		tmp = (((double) M_PI) * (b * b)) * (angle * 0.011111111111111112);
	} else {
		tmp = angle * (((double) M_PI) * (a * (a * -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 (a <= 5e+49) {
		tmp = (Math.PI * (b * b)) * (angle * 0.011111111111111112);
	} else {
		tmp = angle * (Math.PI * (a * (a * -0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 5e+49:
		tmp = (math.pi * (b * b)) * (angle * 0.011111111111111112)
	else:
		tmp = angle * (math.pi * (a * (a * -0.011111111111111112)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 5e+49)
		tmp = Float64(Float64(pi * Float64(b * b)) * Float64(angle * 0.011111111111111112));
	else
		tmp = Float64(angle * Float64(pi * Float64(a * Float64(a * -0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5e+49)
		tmp = (pi * (b * b)) * (angle * 0.011111111111111112);
	else
		tmp = angle * (pi * (a * (a * -0.011111111111111112)));
	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+49], N[(N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(angle * N[(Pi * N[(a * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0000000000000004e49

   1. Initial program 62.1%

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

      1. associate-*l* 62.1%

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

      2. unpow2 62.1%

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

      3. unpow2 62.1%

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

      4. difference-of-squares 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in angle around 0 60.2%

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

   5. Taylor expanded in b around inf 45.6%

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

      1. *-commutative 45.6%

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

      2. *-commutative 45.6%

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

      3. associate-*l* 45.6%

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

      4. *-commutative 45.6%

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

      5. unpow2 45.6%

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

   7. Simplified 45.6%

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

   if 5.0000000000000004e49 < a

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. unpow2 42.7%

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

      3. unpow2 42.7%

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

      4. difference-of-squares 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in angle around 0 59.1%

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

   5. Taylor expanded in b around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l* 50.6%

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

      3. *-commutative 50.6%

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

      4. unpow2 50.6%

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

   7. Simplified 50.6%

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

   8. Taylor expanded in angle around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. unpow2 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*r* 50.6%

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

      5. associate-*r* 50.7%

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

      6. associate-*l* 50.6%

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

      7. *-commutative 50.6%

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

   10. Simplified 50.6%

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

4. Final simplification 46.7%

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

Alternative 15: 47.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.62 \cdot 10^{+50}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= a 1.62e+50)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (* angle (* PI (* a (* a -0.011111111111111112))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.62e+50) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else {
		tmp = angle * (((double) M_PI) * (a * (a * -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 (a <= 1.62e+50) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else {
		tmp = angle * (Math.PI * (a * (a * -0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 1.62e+50:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	else:
		tmp = angle * (math.pi * (a * (a * -0.011111111111111112)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.62e+50)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	else
		tmp = Float64(angle * Float64(pi * Float64(a * Float64(a * -0.011111111111111112))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.62e+50)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	else
		tmp = angle * (pi * (a * (a * -0.011111111111111112)));
	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, 1.62e+50], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(angle * N[(Pi * N[(a * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.61999999999999996e50

   1. Initial program 62.1%

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

      1. associate-*l* 62.1%

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

      2. unpow2 62.1%

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

      3. unpow2 62.1%

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

      4. difference-of-squares 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in angle around 0 60.2%

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

   5. Taylor expanded in b around inf 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-*r* 45.6%

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

      3. unpow2 45.6%

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

   7. Simplified 45.6%

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

   if 1.61999999999999996e50 < a

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. unpow2 42.7%

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

      3. unpow2 42.7%

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

      4. difference-of-squares 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in angle around 0 59.1%

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

   5. Taylor expanded in b around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l* 50.6%

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

      3. *-commutative 50.6%

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

      4. unpow2 50.6%

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

   7. Simplified 50.6%

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

   8. Taylor expanded in angle around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. unpow2 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*r* 50.6%

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

      5. associate-*r* 50.7%

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

      6. associate-*l* 50.6%

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

      7. *-commutative 50.6%

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

   10. Simplified 50.6%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.62 \cdot 10^{+50}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 16: 34.9% accurate, 5.7× speedup

Math

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

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
 (* angle (* PI (* a (* a -0.011111111111111112)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = angle * (pi * (a * (a * -0.011111111111111112)));
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[(angle * N[(Pi * N[(a * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. unpow2 57.8%

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

   3. unpow2 57.8%

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

   4. difference-of-squares 62.5%

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

3. Simplified 62.5%

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

4. Taylor expanded in angle around 0 59.9%

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

5. Taylor expanded in b around 0 37.0%

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

   1. *-commutative 37.0%

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

   2. associate-*l* 37.0%

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

   3. *-commutative 37.0%

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

   4. unpow2 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in angle around 0 37.0%

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

   1. *-commutative 37.0%

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

   2. unpow2 37.0%

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

   3. *-commutative 37.0%

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

   4. associate-*r* 37.0%

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

   5. associate-*r* 37.0%

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

   6. associate-*l* 37.0%

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

   7. *-commutative 37.0%

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

10. Simplified 37.0%

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

11. Final simplification 37.0%

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

Alternative 17: 34.9% accurate, 5.7× speedup

Math

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

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
 (* angle (* -0.011111111111111112 (* PI (* a a)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = angle * (-0.011111111111111112 * (pi * (a * a)));
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[(angle * N[(-0.011111111111111112 * N[(Pi * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. unpow2 57.8%

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

   3. unpow2 57.8%

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

   4. difference-of-squares 62.5%

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

3. Simplified 62.5%

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

4. Taylor expanded in angle around 0 59.9%

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

5. Taylor expanded in b around 0 37.0%

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

   1. *-commutative 37.0%

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

   2. associate-*l* 37.0%

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

   3. *-commutative 37.0%

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

   4. unpow2 37.0%

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

7. Simplified 37.0%

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

8. Final simplification 37.0%

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

Reproduce

herbie shell --seed 2023240 
(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)))))