ab-angle->ABCF B

Percentage Accurate: 54.1% → 67.0%

Time: 22.8s

Alternatives: 21

Speedup: 5.4×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\left(b + a\right) \cdot \left|\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot {\left(\frac{1}{angle}\right)}^{-1}\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) -5e+151)
   (* (+ b a) (fabs (* (- b a) (sin (* angle (* PI 0.011111111111111112))))))
   (if (<= (/ angle 180.0) -2e+21)
     (*
      (+ b a)
      (*
       (- b a)
       (sin (* 2.0 (* PI (* 0.005555555555555556 (pow (/ 1.0 angle) -1.0)))))))
     (if (<= (/ angle 180.0) 3e+112)
       (*
        (+ b a)
        (*
         (- b a)
         (sin (* 2.0 (* (* 0.005555555555555556 angle) (cbrt (pow PI 3.0)))))))
       (* (sin (* 2.0 (* angle (/ PI 180.0)))) (* (+ b a) (- b a)))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= -5e+151) {
		tmp = (b + a) * fabs(((b - a) * sin((angle * (((double) M_PI) * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+21) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (((double) M_PI) * (0.005555555555555556 * pow((1.0 / angle), -1.0))))));
	} else if ((angle / 180.0) <= 3e+112) {
		tmp = (b + a) * ((b - a) * sin((2.0 * ((0.005555555555555556 * angle) * cbrt(pow(((double) M_PI), 3.0))))));
	} else {
		tmp = sin((2.0 * (angle * (((double) M_PI) / 180.0)))) * ((b + a) * (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 ((angle / 180.0) <= -5e+151) {
		tmp = (b + a) * Math.abs(((b - a) * Math.sin((angle * (Math.PI * 0.011111111111111112)))));
	} else if ((angle / 180.0) <= -2e+21) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.PI * (0.005555555555555556 * Math.pow((1.0 / angle), -1.0))))));
	} else if ((angle / 180.0) <= 3e+112) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * ((0.005555555555555556 * angle) * Math.cbrt(Math.pow(Math.PI, 3.0))))));
	} else {
		tmp = Math.sin((2.0 * (angle * (Math.PI / 180.0)))) * ((b + a) * (b - a));
	}
	return tmp;
}

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e+151)
		tmp = Float64(Float64(b + a) * abs(Float64(Float64(b - a) * sin(Float64(angle * Float64(pi * 0.011111111111111112))))));
	elseif (Float64(angle / 180.0) <= -2e+21)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * (Float64(1.0 / angle) ^ -1.0)))))));
	elseif (Float64(angle / 180.0) <= 3e+112)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle) * cbrt((pi ^ 3.0)))))));
	else
		tmp = Float64(sin(Float64(2.0 * Float64(angle * Float64(pi / 180.0)))) * Float64(Float64(b + a) * Float64(b - a)));
	end
	return tmp
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e+151], N[(N[(b + a), $MachinePrecision] * N[Abs[N[(N[(b - a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e+21], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(0.005555555555555556 * N[Power[N[(1.0 / angle), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 3e+112], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(2.0 * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 angle 180) < -5.0000000000000002e151

   1. Initial program 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-*l* 31.0%

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

      3. associate-*l* 31.0%

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

      4. unpow2 31.0%

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

      5. unpow2 31.0%

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

      6. difference-of-squares 31.0%

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

   3. Simplified 31.0%

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

      1. difference-of-squares 31.0%

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

      2. *-commutative 31.0%

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

      3. prod-diff 18.9%

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

      4. fma-neg 18.9%

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

      5. distribute-lft-in 18.9%

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

   5. Applied egg-rr 19.4%

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

      1. *-commutative 19.4%

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

      2. distribute-rgt-out 19.4%

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

      3. *-commutative 19.4%

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

   7. Simplified 34.6%

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

      1. metadata-eval 34.6%

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

      2. div-inv 31.0%

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

      3. clear-num 34.4%

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

   9. Applied egg-rr 34.4%

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

       1. add-sqr-sqrt 18.9%

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

       2. sqrt-unprod 30.8%

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

       3. pow2 30.8%

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

       4. *-commutative 30.8%

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

       5. *-commutative 30.8%

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

       6. associate-/r/ 30.6%

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

       7. metadata-eval 30.6%

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

       8. *-commutative 30.6%

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

       9. associate-*r* 30.5%

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

       10. associate-*l* 30.5%

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

       11. metadata-eval 30.5%

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

   11. Applied egg-rr 30.5%

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

       1. unpow2 30.5%

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

       2. rem-sqrt-square 30.5%

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

       3. *-commutative 30.5%

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

       4. *-commutative 30.5%

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

       5. associate-*l* 30.6%

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

   13. Simplified 30.6%

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

   if -5.0000000000000002e151 < (/.f64 angle 180) < -2e21

   1. Initial program 25.7%

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

      1. *-commutative 25.7%

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

      2. associate-*l* 25.7%

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

      3. associate-*l* 25.7%

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

      4. unpow2 25.7%

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

      5. unpow2 25.7%

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

      6. difference-of-squares 29.1%

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

   3. Simplified 29.1%

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

      1. difference-of-squares 25.7%

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

      2. *-commutative 25.7%

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

      3. prod-diff 15.3%

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

      4. fma-neg 15.3%

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

      5. distribute-lft-in 15.3%

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

   5. Applied egg-rr 15.9%

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

      1. *-commutative 15.9%

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

      2. distribute-rgt-out 15.9%

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

      3. *-commutative 15.9%

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

   7. Simplified 29.7%

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

      1. metadata-eval 29.7%

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

      2. div-inv 29.1%

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

      3. add-cube-cbrt 32.7%

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

      4. pow3 30.9%

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

      5. div-inv 31.1%

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

      6. metadata-eval 31.1%

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

   9. Applied egg-rr 31.1%

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

       1. rem-cube-cbrt 29.7%

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

       2. metadata-eval 29.7%

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

       3. div-inv 29.1%

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

       4. clear-num 31.1%

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

       5. inv-pow 31.1%

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

       6. div-inv 26.1%

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

       7. unpow-prod-down 36.7%

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

       8. metadata-eval 36.7%

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

   11. Applied egg-rr 36.7%

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

   if -2e21 < (/.f64 angle 180) < 2.99999999999999979e112

   1. Initial program 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-*l* 68.5%

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

      3. associate-*l* 68.5%

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

      4. unpow2 68.5%

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

      5. unpow2 68.5%

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

      6. difference-of-squares 69.9%

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

   3. Simplified 69.9%

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

      1. difference-of-squares 68.5%

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

      2. *-commutative 68.5%

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

      3. prod-diff 59.8%

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

      4. fma-neg 59.8%

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

      5. distribute-lft-in 59.8%

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

   5. Applied egg-rr 61.1%

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

      1. *-commutative 61.1%

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

      2. distribute-rgt-out 61.1%

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

      3. *-commutative 61.1%

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

   7. Simplified 86.0%

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

      1. add-cbrt-cube 90.3%

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

      2. pow3 90.3%

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

   9. Applied egg-rr 90.3%

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

   if 2.99999999999999979e112 < (/.f64 angle 180)

   1. Initial program 54.0%

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

      1. *-commutative 54.0%

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

      2. associate-*l* 54.0%

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

      3. associate-*l* 54.0%

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

      4. unpow2 54.0%

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

      5. unpow2 54.0%

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

      6. difference-of-squares 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in angle around inf 47.6%

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

   6. Applied egg-rr 37.4%

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

      1. expm1-def 39.5%

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

      2. expm1-log1p 56.8%

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

      3. *-commutative 56.8%

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

      4. +-commutative 56.8%

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

      5. *-commutative 56.8%

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

      6. metadata-eval 56.8%

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

      7. associate-/r/ 56.8%

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

      8. associate-*r/ 51.8%

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

      9. *-rgt-identity 51.8%

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

      10. associate-/r/ 59.4%

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

   8. Simplified 59.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\frac{\pi}{180} \cdot angle\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\left(b + a\right) \cdot \left|\left(b - a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot {\left(\frac{1}{angle}\right)}^{-1}\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \]

Alternative 2: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 10^{-221}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot {\left(\frac{1}{angle}\right)}^{-1}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left({\left(\sqrt{\pi}\right)}^{2} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (- (pow b 2.0) (pow a 2.0)) 1e-221)
   (*
    (+ b a)
    (*
     (- b a)
     (sin (* 2.0 (* PI (* 0.005555555555555556 (pow (/ 1.0 angle) -1.0)))))))
   (*
    (+ b a)
    (*
     (- b a)
     (sin (* 2.0 (* (pow (sqrt PI) 2.0) (* 0.005555555555555556 angle))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= 1e-221) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (((double) M_PI) * (0.005555555555555556 * pow((1.0 / angle), -1.0))))));
	} else {
		tmp = (b + a) * ((b - a) * sin((2.0 * (pow(sqrt(((double) M_PI)), 2.0) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= 1e-221) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.PI * (0.005555555555555556 * Math.pow((1.0 / angle), -1.0))))));
	} else {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.pow(Math.sqrt(Math.PI), 2.0) * (0.005555555555555556 * angle)))));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= 1e-221:
		tmp = (b + a) * ((b - a) * math.sin((2.0 * (math.pi * (0.005555555555555556 * math.pow((1.0 / angle), -1.0))))))
	else:
		tmp = (b + a) * ((b - a) * math.sin((2.0 * (math.pow(math.sqrt(math.pi), 2.0) * (0.005555555555555556 * angle)))))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= 1e-221)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * (Float64(1.0 / angle) ^ -1.0)))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64((sqrt(pi) ^ 2.0) * Float64(0.005555555555555556 * angle))))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= 1e-221)
		tmp = (b + a) * ((b - a) * sin((2.0 * (pi * (0.005555555555555556 * ((1.0 / angle) ^ -1.0))))));
	else
		tmp = (b + a) * ((b - a) * sin((2.0 * ((sqrt(pi) ^ 2.0) * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], 1e-221], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(0.005555555555555556 * N[Power[N[(1.0 / angle), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 1.00000000000000002e-221

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*l* 65.8%

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

      3. associate-*l* 65.8%

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

      4. unpow2 65.8%

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

      5. unpow2 65.8%

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

      6. difference-of-squares 65.8%

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

   3. Simplified 65.8%

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

      1. difference-of-squares 65.8%

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

      2. *-commutative 65.8%

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

      3. prod-diff 45.3%

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

      4. fma-neg 45.3%

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

      5. distribute-lft-in 45.3%

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

   5. Applied egg-rr 45.6%

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

      1. *-commutative 45.6%

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

      2. distribute-rgt-out 45.5%

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

      3. *-commutative 45.5%

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

   7. Simplified 73.3%

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

      1. metadata-eval 73.3%

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

      2. div-inv 73.0%

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

      3. add-cube-cbrt 70.3%

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

      4. pow3 70.0%

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

      5. div-inv 70.7%

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

      6. metadata-eval 70.7%

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

   9. Applied egg-rr 70.7%

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

       1. rem-cube-cbrt 73.3%

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

       2. metadata-eval 73.3%

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

       3. div-inv 73.0%

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

       4. clear-num 73.4%

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

       5. inv-pow 73.4%

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

       6. div-inv 72.1%

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

       7. unpow-prod-down 73.8%

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

       8. metadata-eval 73.8%

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

   11. Applied egg-rr 73.8%

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

   if 1.00000000000000002e-221 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

      2. associate-*l* 44.8%

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

      3. associate-*l* 44.8%

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

      4. unpow2 44.8%

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

      5. unpow2 44.8%

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

      6. difference-of-squares 48.7%

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

   3. Simplified 48.7%

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

      1. difference-of-squares 44.8%

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

      2. *-commutative 44.8%

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

      3. prod-diff 44.8%

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

      4. fma-neg 44.8%

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

      5. distribute-lft-in 44.8%

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

   5. Applied egg-rr 46.7%

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

      1. *-commutative 46.7%

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

      2. distribute-rgt-out 46.7%

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

      3. *-commutative 46.7%

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

   7. Simplified 62.7%

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

      1. add-sqr-sqrt 70.2%

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

      2. pow2 70.2%

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

   9. Applied egg-rr 70.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 10^{-221}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot {\left(\frac{1}{angle}\right)}^{-1}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left({\left(\sqrt{\pi}\right)}^{2} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{\pi \cdot angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 3e+112)
   (* (+ b a) (* (- b a) (sin (* 2.0 (/ (* PI angle) 180.0)))))
   (* (sin (* 2.0 (* angle (/ PI 180.0)))) (* (+ b a) (- b a)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 3e+112) {
		tmp = (b + a) * ((b - a) * sin((2.0 * ((((double) M_PI) * angle) / 180.0))));
	} else {
		tmp = sin((2.0 * (angle * (((double) M_PI) / 180.0)))) * ((b + a) * (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 ((angle / 180.0) <= 3e+112) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * ((Math.PI * angle) / 180.0))));
	} else {
		tmp = Math.sin((2.0 * (angle * (Math.PI / 180.0)))) * ((b + a) * (b - a));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if (angle / 180.0) <= 3e+112:
		tmp = (b + a) * ((b - a) * math.sin((2.0 * ((math.pi * angle) / 180.0))))
	else:
		tmp = math.sin((2.0 * (angle * (math.pi / 180.0)))) * ((b + a) * (b - a))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 3e+112)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(pi * angle) / 180.0)))));
	else
		tmp = Float64(sin(Float64(2.0 * Float64(angle * Float64(pi / 180.0)))) * Float64(Float64(b + a) * 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 ((angle / 180.0) <= 3e+112)
		tmp = (b + a) * ((b - a) * sin((2.0 * ((pi * angle) / 180.0))));
	else
		tmp = sin((2.0 * (angle * (pi / 180.0)))) * ((b + a) * (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[(angle / 180.0), $MachinePrecision], 3e+112], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(2.0 * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle 180) < 2.99999999999999979e112

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-*l* 57.1%

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

      3. associate-*l* 57.1%

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

      4. unpow2 57.1%

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

      5. unpow2 57.1%

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

      6. difference-of-squares 58.6%

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

   3. Simplified 58.6%

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

      1. difference-of-squares 57.1%

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

      2. *-commutative 57.1%

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

      3. prod-diff 47.7%

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

      4. fma-neg 47.7%

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

      5. distribute-lft-in 47.7%

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

   5. Applied egg-rr 48.8%

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

      1. *-commutative 48.8%

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

      2. distribute-rgt-out 48.7%

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

      3. *-commutative 48.7%

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

   7. Simplified 70.7%

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

      1. metadata-eval 70.7%

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

      2. div-inv 70.5%

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

      3. associate-*r/ 72.3%

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

   9. Applied egg-rr 72.3%

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

   if 2.99999999999999979e112 < (/.f64 angle 180)

   1. Initial program 54.0%

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

      1. *-commutative 54.0%

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

      2. associate-*l* 54.0%

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

      3. associate-*l* 54.0%

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

      4. unpow2 54.0%

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

      5. unpow2 54.0%

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

      6. difference-of-squares 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in angle around inf 47.6%

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

   6. Applied egg-rr 37.4%

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

      1. expm1-def 39.5%

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

      2. expm1-log1p 56.8%

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

      3. *-commutative 56.8%

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

      4. +-commutative 56.8%

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

      5. *-commutative 56.8%

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

      6. metadata-eval 56.8%

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

      7. associate-/r/ 56.8%

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

      8. associate-*r/ 51.8%

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

      9. *-rgt-identity 51.8%

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

      10. associate-/r/ 59.4%

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

   8. Simplified 59.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{\pi \cdot angle}{180}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\\ \end{array} \]

Alternative 4: 66.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+189}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

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 3.6e+189)
   (* (+ b a) (* (- b a) (sin (* 2.0 (/ PI (/ 180.0 angle))))))
   (* (+ b a) (* (sin (* PI (* angle 0.011111111111111112))) (- a)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.6e+189) {
		tmp = (b + a) * ((b - a) * sin((2.0 * (((double) M_PI) / (180.0 / angle)))));
	} else {
		tmp = (b + a) * (sin((((double) M_PI) * (angle * 0.011111111111111112))) * -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 <= 3.6e+189) {
		tmp = (b + a) * ((b - a) * Math.sin((2.0 * (Math.PI / (180.0 / angle)))));
	} else {
		tmp = (b + a) * (Math.sin((Math.PI * (angle * 0.011111111111111112))) * -a);
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 3.6e+189:
		tmp = (b + a) * ((b - a) * math.sin((2.0 * (math.pi / (180.0 / angle)))))
	else:
		tmp = (b + a) * (math.sin((math.pi * (angle * 0.011111111111111112))) * -a)
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.6e+189)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi / Float64(180.0 / angle))))));
	else
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(pi * Float64(angle * 0.011111111111111112))) * Float64(-a)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.6e+189)
		tmp = (b + a) * ((b - a) * sin((2.0 * (pi / (180.0 / angle)))));
	else
		tmp = (b + a) * (sin((pi * (angle * 0.011111111111111112))) * -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, 3.6e+189], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.60000000000000008e189

   1. Initial program 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-*l* 57.7%

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

      3. associate-*l* 57.7%

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

      4. unpow2 57.7%

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

      5. unpow2 57.7%

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

      6. difference-of-squares 58.6%

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

   3. Simplified 58.6%

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

      1. difference-of-squares 57.7%

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

      2. *-commutative 57.7%

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

      3. prod-diff 50.2%

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

      4. fma-neg 50.2%

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

      5. distribute-lft-in 50.2%

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

   5. Applied egg-rr 51.3%

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

      1. *-commutative 51.3%

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

      2. distribute-rgt-out 51.2%

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

      3. *-commutative 51.2%

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

   7. Simplified 67.7%

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

      1. metadata-eval 67.7%

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

      2. div-inv 67.5%

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

      3. clear-num 68.3%

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

   9. Applied egg-rr 68.3%

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

       1. un-div-inv 68.3%

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

   11. Applied egg-rr 68.3%

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

   if 3.60000000000000008e189 < a

   1. Initial program 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*l* 47.0%

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

      3. associate-*l* 47.0%

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

      4. unpow2 47.0%

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

      5. unpow2 47.0%

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

      6. difference-of-squares 55.4%

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

   3. Simplified 55.4%

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

      1. difference-of-squares 47.0%

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

      2. *-commutative 47.0%

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

      3. prod-diff 0.0%

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

      4. fma-neg 0.0%

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

      5. distribute-lft-in 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-rgt-out 0.0%

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

      3. *-commutative 0.0%

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

   7. Simplified 76.9%

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

      1. metadata-eval 76.9%

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

      2. div-inv 76.9%

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

      3. clear-num 80.6%

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

   9. Applied egg-rr 80.6%

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

   10. Taylor expanded in b around 0 76.7%

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

       1. mul-1-neg 76.7%

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

       2. *-commutative 76.7%

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

       3. *-commutative 76.7%

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

       4. *-commutative 76.7%

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

       5. distribute-rgt-neg-out 76.7%

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

       6. associate-*l* 80.8%

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

   12. Simplified 80.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+189}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(-a\right)\right)\\ \end{array} \]

Alternative 5: 67.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{1}{\frac{180}{angle}}\right)\right)\right) \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* (+ b a) (* (- b a) (sin (* 2.0 (* PI (/ 1.0 (/ 180.0 angle))))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return (b + a) * ((b - a) * sin((2.0 * (((double) M_PI) * (1.0 / (180.0 / angle))))));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return (b + a) * ((b - a) * Math.sin((2.0 * (Math.PI * (1.0 / (180.0 / angle))))));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return (b + a) * ((b - a) * math.sin((2.0 * (math.pi * (1.0 / (180.0 / angle))))))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * Float64(1.0 / Float64(180.0 / angle)))))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = (b + a) * ((b - a) * sin((2.0 * (pi * (1.0 / (180.0 / angle))))));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(1.0 / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 56.6%

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

   2. associate-*l* 56.6%

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

   3. associate-*l* 56.6%

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

   4. unpow2 56.6%

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

   5. unpow2 56.6%

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

   6. difference-of-squares 58.3%

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

3. Simplified 58.3%

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

   1. difference-of-squares 56.6%

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

   2. *-commutative 56.6%

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

   3. prod-diff 45.1%

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

   4. fma-neg 45.1%

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

   5. distribute-lft-in 45.1%

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

5. Applied egg-rr 46.1%

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

   1. *-commutative 46.1%

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

   2. distribute-rgt-out 46.0%

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

   3. *-commutative 46.0%

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

7. Simplified 68.6%

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

   1. metadata-eval 68.6%

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

   2. div-inv 68.5%

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

   3. clear-num 69.5%

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

9. Applied egg-rr 69.5%

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

10. Final simplification 69.5%

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

Alternative 6: 64.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} t_0 := \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;a \leq 1.5 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot t_0\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+173}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(a \cdot \left(-t_0\right)\right)\\ \end{array} \end{array} \]

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 (sin (* angle (* PI 0.011111111111111112)))))
   (if (<= a 1.5e-101)
     (* (+ b a) (* b t_0))
     (if (<= a 3.6e+173)
       (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
       (* (+ b a) (* a (- t_0)))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = sin((angle * (((double) M_PI) * 0.011111111111111112)));
	double tmp;
	if (a <= 1.5e-101) {
		tmp = (b + a) * (b * t_0);
	} else if (a <= 3.6e+173) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = (b + a) * (a * -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.sin((angle * (Math.PI * 0.011111111111111112)));
	double tmp;
	if (a <= 1.5e-101) {
		tmp = (b + a) * (b * t_0);
	} else if (a <= 3.6e+173) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = (b + a) * (a * -t_0);
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	t_0 = math.sin((angle * (math.pi * 0.011111111111111112)))
	tmp = 0
	if a <= 1.5e-101:
		tmp = (b + a) * (b * t_0)
	elif a <= 3.6e+173:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = (b + a) * (a * -t_0)
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	t_0 = sin(Float64(angle * Float64(pi * 0.011111111111111112)))
	tmp = 0.0
	if (a <= 1.5e-101)
		tmp = Float64(Float64(b + a) * Float64(b * t_0));
	elseif (a <= 3.6e+173)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(Float64(b + a) * Float64(a * Float64(-t_0)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = sin((angle * (pi * 0.011111111111111112)));
	tmp = 0.0;
	if (a <= 1.5e-101)
		tmp = (b + a) * (b * t_0);
	elseif (a <= 3.6e+173)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = (b + a) * (a * -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[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, 1.5e-101], N[(N[(b + a), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e+173], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(a * (-t$95$0)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.5000000000000002e-101

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. associate-*l* 57.6%

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

      4. unpow2 57.6%

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

      5. unpow2 57.6%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

      1. difference-of-squares 57.6%

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

      2. *-commutative 57.6%

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

      3. prod-diff 49.8%

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

      4. fma-neg 49.8%

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

      5. distribute-lft-in 49.9%

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

   5. Applied egg-rr 50.5%

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

      1. *-commutative 50.5%

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

      2. distribute-rgt-out 50.5%

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

      3. *-commutative 50.5%

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

   7. Simplified 65.4%

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

      1. metadata-eval 65.4%

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

      2. div-inv 65.4%

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

      3. clear-num 65.8%

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

   9. Applied egg-rr 65.8%

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

   10. Taylor expanded in b around inf 44.6%

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

       1. *-commutative 44.6%

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

       2. associate-*l* 46.3%

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

   12. Simplified 46.3%

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

   if 1.5000000000000002e-101 < a < 3.6000000000000002e173

   1. Initial program 58.6%

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

      1. *-commutative 58.6%

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

      2. associate-*l* 58.6%

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

      3. associate-*l* 58.6%

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

      4. unpow2 58.6%

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

      5. unpow2 58.6%

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

      6. difference-of-squares 58.6%

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

   3. Simplified 58.6%

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

      1. difference-of-squares 58.6%

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

      2. *-commutative 58.6%

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

      3. prod-diff 55.3%

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

      4. fma-neg 55.3%

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

      5. distribute-lft-in 55.4%

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

   5. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. distribute-rgt-out 55.9%

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

      3. *-commutative 55.9%

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

   7. Simplified 71.6%

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

      1. metadata-eval 71.6%

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

      2. div-inv 71.0%

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

      3. clear-num 72.7%

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

   9. Applied egg-rr 72.7%

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

   10. Taylor expanded in angle around 0 67.5%

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

       1. associate-*r* 67.5%

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

   12. Simplified 67.5%

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

   if 3.6000000000000002e173 < a

   1. Initial program 47.6%

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

      1. *-commutative 47.6%

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

      2. associate-*l* 47.6%

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

      3. associate-*l* 47.6%

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

      4. unpow2 47.6%

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

      5. unpow2 47.6%

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

      6. difference-of-squares 57.7%

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

   3. Simplified 57.7%

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

      1. difference-of-squares 47.6%

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

      2. *-commutative 47.6%

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

      3. prod-diff 0.0%

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

      4. fma-neg 0.0%

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

      5. distribute-lft-in 0.0%

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

   5. Applied egg-rr 3.1%

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

      1. *-commutative 3.1%

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

      2. distribute-rgt-out 3.1%

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

      3. *-commutative 3.1%

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

   7. Simplified 78.1%

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

      1. metadata-eval 78.1%

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

      2. div-inv 78.1%

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

      3. clear-num 81.2%

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

   9. Applied egg-rr 81.2%

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

   10. Taylor expanded in b around 0 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. *-commutative 75.0%

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

       4. *-commutative 75.0%

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

       5. distribute-rgt-neg-out 75.0%

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

       6. *-commutative 75.0%

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

       7. associate-*l* 75.1%

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

   12. Simplified 75.1%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+173}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(a \cdot \left(-\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+173}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(-a\right)\right)\\ \end{array} \end{array} \]

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 2.2e-101)
   (* (+ b a) (* b (sin (* angle (* PI 0.011111111111111112)))))
   (if (<= a 2.5e+173)
     (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
     (* (+ b a) (* (sin (* PI (* angle 0.011111111111111112))) (- a))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.2e-101) {
		tmp = (b + a) * (b * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else if (a <= 2.5e+173) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = (b + a) * (sin((((double) M_PI) * (angle * 0.011111111111111112))) * -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 <= 2.2e-101) {
		tmp = (b + a) * (b * Math.sin((angle * (Math.PI * 0.011111111111111112))));
	} else if (a <= 2.5e+173) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = (b + a) * (Math.sin((Math.PI * (angle * 0.011111111111111112))) * -a);
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 2.2e-101:
		tmp = (b + a) * (b * math.sin((angle * (math.pi * 0.011111111111111112))))
	elif a <= 2.5e+173:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = (b + a) * (math.sin((math.pi * (angle * 0.011111111111111112))) * -a)
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.2e-101)
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	elseif (a <= 2.5e+173)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(pi * Float64(angle * 0.011111111111111112))) * Float64(-a)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.2e-101)
		tmp = (b + a) * (b * sin((angle * (pi * 0.011111111111111112))));
	elseif (a <= 2.5e+173)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = (b + a) * (sin((pi * (angle * 0.011111111111111112))) * -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, 2.2e-101], N[(N[(b + a), $MachinePrecision] * N[(b * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.5e+173], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.1999999999999999e-101

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. associate-*l* 57.6%

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

      4. unpow2 57.6%

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

      5. unpow2 57.6%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

      1. difference-of-squares 57.6%

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

      2. *-commutative 57.6%

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

      3. prod-diff 49.8%

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

      4. fma-neg 49.8%

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

      5. distribute-lft-in 49.9%

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

   5. Applied egg-rr 50.5%

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

      1. *-commutative 50.5%

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

      2. distribute-rgt-out 50.5%

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

      3. *-commutative 50.5%

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

   7. Simplified 65.4%

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

      1. metadata-eval 65.4%

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

      2. div-inv 65.4%

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

      3. clear-num 65.8%

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

   9. Applied egg-rr 65.8%

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

   10. Taylor expanded in b around inf 44.6%

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

       1. *-commutative 44.6%

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

       2. associate-*l* 46.3%

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

   12. Simplified 46.3%

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

   if 2.1999999999999999e-101 < a < 2.50000000000000017e173

   1. Initial program 58.6%

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

      1. *-commutative 58.6%

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

      2. associate-*l* 58.6%

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

      3. associate-*l* 58.6%

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

      4. unpow2 58.6%

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

      5. unpow2 58.6%

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

      6. difference-of-squares 58.6%

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

   3. Simplified 58.6%

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

      1. difference-of-squares 58.6%

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

      2. *-commutative 58.6%

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

      3. prod-diff 55.3%

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

      4. fma-neg 55.3%

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

      5. distribute-lft-in 55.4%

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

   5. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. distribute-rgt-out 55.9%

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

      3. *-commutative 55.9%

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

   7. Simplified 71.6%

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

      1. metadata-eval 71.6%

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

      2. div-inv 71.0%

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

      3. clear-num 72.7%

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

   9. Applied egg-rr 72.7%

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

   10. Taylor expanded in angle around 0 67.5%

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

       1. associate-*r* 67.5%

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

   12. Simplified 67.5%

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

   if 2.50000000000000017e173 < a

   1. Initial program 47.6%

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

      1. *-commutative 47.6%

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

      2. associate-*l* 47.6%

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

      3. associate-*l* 47.6%

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

      4. unpow2 47.6%

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

      5. unpow2 47.6%

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

      6. difference-of-squares 57.7%

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

   3. Simplified 57.7%

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

      1. difference-of-squares 47.6%

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

      2. *-commutative 47.6%

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

      3. prod-diff 0.0%

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

      4. fma-neg 0.0%

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

      5. distribute-lft-in 0.0%

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

   5. Applied egg-rr 3.1%

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

      1. *-commutative 3.1%

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

      2. distribute-rgt-out 3.1%

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

      3. *-commutative 3.1%

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

   7. Simplified 78.1%

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

      1. metadata-eval 78.1%

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

      2. div-inv 78.1%

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

      3. clear-num 81.2%

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

   9. Applied egg-rr 81.2%

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

   10. Taylor expanded in b around 0 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. *-commutative 75.0%

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

       4. *-commutative 75.0%

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

       5. distribute-rgt-neg-out 75.0%

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

       6. associate-*l* 78.3%

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

   12. Simplified 78.3%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+173}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(-a\right)\right)\\ \end{array} \]

Alternative 8: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 9.8 \cdot 10^{-128}:\\ \;\;\;\;\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 9.8e-128)
   (* (* 2.0 (* b b)) (sin (* PI (* 0.005555555555555556 angle))))
   (* (+ b a) (* (- b a) (* PI (* angle 0.011111111111111112))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9.8e-128) {
		tmp = (2.0 * (b * b)) * sin((((double) M_PI) * (0.005555555555555556 * angle)));
	} else {
		tmp = (b + a) * ((b - a) * (((double) M_PI) * (angle * 0.011111111111111112)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9.8e-128) {
		tmp = (2.0 * (b * b)) * Math.sin((Math.PI * (0.005555555555555556 * angle)));
	} else {
		tmp = (b + a) * ((b - a) * (Math.PI * (angle * 0.011111111111111112)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 9.8e-128:
		tmp = (2.0 * (b * b)) * math.sin((math.pi * (0.005555555555555556 * angle)))
	else:
		tmp = (b + a) * ((b - a) * (math.pi * (angle * 0.011111111111111112)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 9.8e-128)
		tmp = Float64(Float64(2.0 * Float64(b * b)) * sin(Float64(pi * Float64(0.005555555555555556 * angle))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(pi * 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 (a <= 9.8e-128)
		tmp = (2.0 * (b * b)) * sin((pi * (0.005555555555555556 * angle)));
	else
		tmp = (b + a) * ((b - a) * (pi * (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[a, 9.8e-128], N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.79999999999999979e-128

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

      2. associate-*l* 57.5%

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

      3. associate-*l* 57.5%

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

      4. unpow2 57.5%

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

      5. unpow2 57.5%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in angle around inf 58.7%

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

   5. Taylor expanded in angle around 0 58.3%

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

   6. Taylor expanded in b around inf 38.7%

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

      1. associate-*r* 38.7%

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

      2. unpow2 38.7%

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

      3. associate-*r* 38.8%

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

      4. *-commutative 38.8%

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

      5. *-commutative 38.8%

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

   8. Simplified 38.8%

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

   if 9.79999999999999979e-128 < a

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*l* 55.4%

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

      3. associate-*l* 55.4%

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

      4. unpow2 55.4%

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

      5. unpow2 55.4%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

      1. difference-of-squares 55.4%

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

      2. *-commutative 55.4%

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

      3. prod-diff 39.2%

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

      4. fma-neg 39.2%

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

      5. distribute-lft-in 39.2%

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

   5. Applied egg-rr 40.5%

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

      1. *-commutative 40.5%

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

      2. distribute-rgt-out 40.5%

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

      3. *-commutative 40.5%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in angle around 0 66.6%

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

      1. associate-*r* 66.6%

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

   10. Simplified 66.6%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.8 \cdot 10^{-128}:\\ \;\;\;\;\left(2 \cdot \left(b \cdot b\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 9: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{-97}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

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 1e-97)
   (* (+ b a) (* b (sin (* 0.011111111111111112 (* PI angle)))))
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e-97) {
		tmp = (b + a) * (b * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e-97) {
		tmp = (b + a) * (b * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 1e-97:
		tmp = (b + a) * (b * math.sin((0.011111111111111112 * (math.pi * angle))))
	else:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1e-97)
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1e-97)
		tmp = (b + a) * (b * sin((0.011111111111111112 * (pi * angle))));
	else
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1e-97], N[(N[(b + a), $MachinePrecision] * N[(b * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.00000000000000004e-97

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. associate-*l* 57.6%

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

      4. unpow2 57.6%

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

      5. unpow2 57.6%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

      1. difference-of-squares 57.6%

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

      2. *-commutative 57.6%

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

      3. prod-diff 49.8%

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

      4. fma-neg 49.8%

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

      5. distribute-lft-in 49.9%

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

   5. Applied egg-rr 50.5%

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

      1. *-commutative 50.5%

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

      2. distribute-rgt-out 50.5%

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

      3. *-commutative 50.5%

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

   7. Simplified 65.4%

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

      1. metadata-eval 65.4%

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

      2. div-inv 65.4%

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

      3. clear-num 65.8%

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

   9. Applied egg-rr 65.8%

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

   10. Taylor expanded in b around inf 44.6%

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

   if 1.00000000000000004e-97 < a

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 55.1%

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

      3. associate-*l* 55.1%

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

      4. unpow2 55.1%

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

      5. unpow2 55.1%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

      1. difference-of-squares 55.1%

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

      2. *-commutative 55.1%

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

      3. prod-diff 37.6%

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

      4. fma-neg 37.6%

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

      5. distribute-lft-in 37.7%

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

   5. Applied egg-rr 39.1%

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

      1. *-commutative 39.1%

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

      2. distribute-rgt-out 39.0%

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

      3. *-commutative 39.0%

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

   7. Simplified 73.7%

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

      1. metadata-eval 73.7%

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

      2. div-inv 73.3%

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

      3. clear-num 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in angle around 0 67.8%

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

       1. associate-*r* 67.9%

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

   12. Simplified 67.9%

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

4. Final simplification 53.7%

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

Alternative 10: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

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 4.5e-101)
   (* (+ b a) (* b (sin (* angle (* PI 0.011111111111111112)))))
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.5e-101) {
		tmp = (b + a) * (b * sin((angle * (((double) M_PI) * 0.011111111111111112))));
	} else {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.5e-101) {
		tmp = (b + a) * (b * Math.sin((angle * (Math.PI * 0.011111111111111112))));
	} else {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if a <= 4.5e-101:
		tmp = (b + a) * (b * math.sin((angle * (math.pi * 0.011111111111111112))))
	else:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.5e-101)
		tmp = Float64(Float64(b + a) * Float64(b * sin(Float64(angle * Float64(pi * 0.011111111111111112)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.5e-101)
		tmp = (b + a) * (b * sin((angle * (pi * 0.011111111111111112))));
	else
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 4.5e-101], N[(N[(b + a), $MachinePrecision] * N[(b * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.4999999999999998e-101

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. associate-*l* 57.6%

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

      4. unpow2 57.6%

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

      5. unpow2 57.6%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

      1. difference-of-squares 57.6%

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

      2. *-commutative 57.6%

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

      3. prod-diff 49.8%

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

      4. fma-neg 49.8%

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

      5. distribute-lft-in 49.9%

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

   5. Applied egg-rr 50.5%

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

      1. *-commutative 50.5%

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

      2. distribute-rgt-out 50.5%

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

      3. *-commutative 50.5%

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

   7. Simplified 65.4%

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

      1. metadata-eval 65.4%

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

      2. div-inv 65.4%

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

      3. clear-num 65.8%

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

   9. Applied egg-rr 65.8%

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

   10. Taylor expanded in b around inf 44.6%

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

       1. *-commutative 44.6%

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

       2. associate-*l* 46.3%

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

   12. Simplified 46.3%

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

   if 4.4999999999999998e-101 < a

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 55.1%

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

      3. associate-*l* 55.1%

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

      4. unpow2 55.1%

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

      5. unpow2 55.1%

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

      6. difference-of-squares 58.3%

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

   3. Simplified 58.3%

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

      1. difference-of-squares 55.1%

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

      2. *-commutative 55.1%

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

      3. prod-diff 37.6%

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

      4. fma-neg 37.6%

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

      5. distribute-lft-in 37.7%

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

   5. Applied egg-rr 39.1%

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

      1. *-commutative 39.1%

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

      2. distribute-rgt-out 39.0%

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

      3. *-commutative 39.0%

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

   7. Simplified 73.7%

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

      1. metadata-eval 73.7%

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

      2. div-inv 73.3%

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

      3. clear-num 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in angle around 0 67.8%

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

       1. associate-*r* 67.9%

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

   12. Simplified 67.9%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{-101}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \end{array} \]

Alternative 11: 67.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. *-commutative 56.6%

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

   2. associate-*l* 56.6%

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

   3. associate-*l* 56.6%

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

   4. unpow2 56.6%

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

   5. unpow2 56.6%

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

   6. difference-of-squares 58.3%

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

3. Simplified 58.3%

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

   1. difference-of-squares 56.6%

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

   2. *-commutative 56.6%

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

   3. prod-diff 45.1%

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

   4. fma-neg 45.1%

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

   5. distribute-lft-in 45.1%

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

5. Applied egg-rr 46.1%

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

   1. *-commutative 46.1%

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

   2. distribute-rgt-out 46.0%

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

   3. *-commutative 46.0%

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

7. Simplified 68.6%

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

8. Taylor expanded in angle around inf 68.5%

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

9. Final simplification 68.5%

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

Alternative 12: 67.3% accurate, 2.9× speedup

Math

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

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
 (* (+ b a) (* (- b a) (sin (* PI (* angle 0.011111111111111112))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

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. *-commutative 56.6%

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

   2. associate-*l* 56.6%

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

   3. associate-*l* 56.6%

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

   4. unpow2 56.6%

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

   5. unpow2 56.6%

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

   6. difference-of-squares 58.3%

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

3. Simplified 58.3%

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

   1. difference-of-squares 56.6%

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

   2. *-commutative 56.6%

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

   3. prod-diff 45.1%

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

   4. fma-neg 45.1%

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

   5. distribute-lft-in 45.1%

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

5. Applied egg-rr 46.1%

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

   1. *-commutative 46.1%

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

   2. distribute-rgt-out 46.0%

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

   3. *-commutative 46.0%

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

7. Simplified 68.6%

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

8. Taylor expanded in angle around inf 68.5%

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

   1. associate-*r* 68.6%

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

10. Simplified 68.6%

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

11. Final simplification 68.6%

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

Alternative 13: 57.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+132}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.8e+132)
   (* 0.011111111111111112 (* angle (* PI (* (+ b a) (- b a)))))
   (* PI (* 0.011111111111111112 (* b (* b angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.8e+132) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * ((b + a) * (b - a))));
	} else {
		tmp = ((double) M_PI) * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.8e+132) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * ((b + a) * (b - a))));
	} else {
		tmp = Math.PI * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 6.8e+132:
		tmp = 0.011111111111111112 * (angle * (math.pi * ((b + a) * (b - a))))
	else:
		tmp = math.pi * (0.011111111111111112 * (b * (b * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.8e+132)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(Float64(b + a) * Float64(b - a)))));
	else
		tmp = Float64(pi * Float64(0.011111111111111112 * Float64(b * Float64(b * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.8e+132)
		tmp = 0.011111111111111112 * (angle * (pi * ((b + a) * (b - a))));
	else
		tmp = pi * (0.011111111111111112 * (b * (b * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 6.8e+132], N[(0.011111111111111112 * N[(angle * N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.011111111111111112 * N[(b * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.80000000000000051e132

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 58.9%

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

      3. associate-*l* 58.9%

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

      4. unpow2 58.9%

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

      5. unpow2 58.9%

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

      6. difference-of-squares 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in angle around 0 54.8%

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

   if 6.80000000000000051e132 < b

   1. Initial program 41.1%

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

      1. *-commutative 41.1%

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

      2. associate-*l* 41.1%

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

      3. associate-*l* 41.1%

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

      4. unpow2 41.1%

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

      5. unpow2 41.1%

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

      6. difference-of-squares 50.8%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in angle around 0 47.8%

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

   5. Taylor expanded in a around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. *-commutative 47.2%

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

      3. unpow2 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in angle around 0 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-*r* 47.2%

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

      3. unpow2 47.2%

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

      4. *-commutative 47.2%

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

      5. associate-*l* 47.2%

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

      6. *-commutative 47.2%

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

      7. associate-*r* 66.6%

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

      8. *-commutative 66.6%

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

   10. Simplified 66.6%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+132}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 14: 61.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

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 1.5e+67)
   (* (+ b a) (* 0.011111111111111112 (* angle (* (- b a) PI))))
   (* (* PI angle) (* -0.011111111111111112 (* a a)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 1.5e+67) {
		tmp = (b + a) * (0.011111111111111112 * (angle * ((b - a) * ((double) M_PI))));
	} else {
		tmp = (((double) M_PI) * angle) * (-0.011111111111111112 * (a * 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 (angle <= 1.5e+67) {
		tmp = (b + a) * (0.011111111111111112 * (angle * ((b - a) * Math.PI)));
	} else {
		tmp = (Math.PI * angle) * (-0.011111111111111112 * (a * a));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 1.5e+67:
		tmp = (b + a) * (0.011111111111111112 * (angle * ((b - a) * math.pi)))
	else:
		tmp = (math.pi * angle) * (-0.011111111111111112 * (a * a))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 1.5e+67)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle * Float64(Float64(b - a) * pi))));
	else
		tmp = Float64(Float64(pi * angle) * Float64(-0.011111111111111112 * Float64(a * a)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 1.5e+67)
		tmp = (b + a) * (0.011111111111111112 * (angle * ((b - a) * pi)));
	else
		tmp = (pi * angle) * (-0.011111111111111112 * (a * 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[angle, 1.5e+67], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * angle), $MachinePrecision] * N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.50000000000000005e67

   1. Initial program 59.9%

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

      1. *-commutative 59.9%

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

      2. associate-*l* 59.9%

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

      3. associate-*l* 59.9%

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

      4. unpow2 59.9%

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

      5. unpow2 59.9%

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

      6. difference-of-squares 61.5%

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

   3. Simplified 61.5%

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

      1. difference-of-squares 59.9%

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

      2. *-commutative 59.9%

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

      3. prod-diff 50.3%

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

      4. fma-neg 50.3%

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

      5. distribute-lft-in 50.3%

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

   5. Applied egg-rr 51.5%

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

      1. *-commutative 51.5%

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

      2. distribute-rgt-out 51.5%

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

      3. *-commutative 51.5%

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

   7. Simplified 74.8%

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

      1. metadata-eval 74.8%

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

      2. div-inv 74.2%

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

      3. clear-num 75.5%

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

   9. Applied egg-rr 75.5%

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

   10. Taylor expanded in angle around 0 71.7%

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

   if 1.50000000000000005e67 < angle

   1. Initial program 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-*l* 43.6%

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

      3. associate-*l* 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

      6. difference-of-squares 45.6%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in angle around 0 28.8%

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

   5. Taylor expanded in a around inf 29.7%

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

      1. associate-*r* 29.7%

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

      2. unpow2 29.7%

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

   7. Simplified 29.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 15: 61.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

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 1.5e+67)
   (* (+ b a) (* (* angle 0.011111111111111112) (* (- b a) PI)))
   (* (* PI angle) (* -0.011111111111111112 (* a a)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 1.5e+67) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) * angle) * (-0.011111111111111112 * (a * 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 (angle <= 1.5e+67) {
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * Math.PI));
	} else {
		tmp = (Math.PI * angle) * (-0.011111111111111112 * (a * a));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if angle <= 1.5e+67:
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * math.pi))
	else:
		tmp = (math.pi * angle) * (-0.011111111111111112 * (a * a))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (angle <= 1.5e+67)
		tmp = Float64(Float64(b + a) * Float64(Float64(angle * 0.011111111111111112) * Float64(Float64(b - a) * pi)));
	else
		tmp = Float64(Float64(pi * angle) * Float64(-0.011111111111111112 * Float64(a * a)));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 1.5e+67)
		tmp = (b + a) * ((angle * 0.011111111111111112) * ((b - a) * pi));
	else
		tmp = (pi * angle) * (-0.011111111111111112 * (a * 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[angle, 1.5e+67], N[(N[(b + a), $MachinePrecision] * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * angle), $MachinePrecision] * N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.50000000000000005e67

   1. Initial program 59.9%

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

      1. *-commutative 59.9%

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

      2. associate-*l* 59.9%

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

      3. associate-*l* 59.9%

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

      4. unpow2 59.9%

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

      5. unpow2 59.9%

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

      6. difference-of-squares 61.5%

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

   3. Simplified 61.5%

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

      1. difference-of-squares 59.9%

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

      2. *-commutative 59.9%

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

      3. prod-diff 50.3%

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

      4. fma-neg 50.3%

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

      5. distribute-lft-in 50.3%

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

   5. Applied egg-rr 51.5%

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

      1. *-commutative 51.5%

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

      2. distribute-rgt-out 51.5%

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

      3. *-commutative 51.5%

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

   7. Simplified 74.8%

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

      1. metadata-eval 74.8%

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

      2. div-inv 74.2%

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

      3. clear-num 75.5%

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

   9. Applied egg-rr 75.5%

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

   10. Taylor expanded in angle around 0 71.7%

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

       1. associate-*r* 71.8%

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

   12. Simplified 71.8%

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

   if 1.50000000000000005e67 < angle

   1. Initial program 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-*l* 43.6%

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

      3. associate-*l* 43.6%

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

      4. unpow2 43.6%

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

      5. unpow2 43.6%

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

      6. difference-of-squares 45.6%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in angle around 0 28.8%

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

   5. Taylor expanded in a around inf 29.7%

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

      1. associate-*r* 29.7%

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

      2. unpow2 29.7%

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

   7. Simplified 29.7%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 16: 50.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+80}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(-a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 7e+80)
   (* 0.011111111111111112 (* angle (* PI (- (* a a)))))
   (* PI (* 0.011111111111111112 (* b (* b angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 7e+80) {
		tmp = 0.011111111111111112 * (angle * (((double) M_PI) * -(a * a)));
	} else {
		tmp = ((double) M_PI) * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 7e+80) {
		tmp = 0.011111111111111112 * (angle * (Math.PI * -(a * a)));
	} else {
		tmp = Math.PI * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 7e+80:
		tmp = 0.011111111111111112 * (angle * (math.pi * -(a * a)))
	else:
		tmp = math.pi * (0.011111111111111112 * (b * (b * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 7e+80)
		tmp = Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(-Float64(a * a)))));
	else
		tmp = Float64(pi * Float64(0.011111111111111112 * Float64(b * Float64(b * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 7e+80)
		tmp = 0.011111111111111112 * (angle * (pi * -(a * a)));
	else
		tmp = pi * (0.011111111111111112 * (b * (b * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 7e+80], N[(0.011111111111111112 * N[(angle * N[(Pi * (-N[(a * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.011111111111111112 * N[(b * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.99999999999999987e80

   1. Initial program 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-*l* 60.0%

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

      3. associate-*l* 60.0%

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

      4. unpow2 60.0%

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

      5. unpow2 60.0%

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

      6. difference-of-squares 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in angle around 0 55.9%

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

   5. Taylor expanded in a around inf 41.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \pi\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 41.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\left(-1 \cdot {a}^{2}\right) \cdot \pi\right)}\right) \]

      2. *-commutative 41.7%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot \left(-1 \cdot {a}^{2}\right)\right)}\right) \]

      3. mul-1-neg 41.7%

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

      4. unpow2 41.7%

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

   7. Simplified 41.7%

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

   if 6.99999999999999987e80 < b

   1. Initial program 39.8%

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

      1. *-commutative 39.8%

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

      2. associate-*l* 39.8%

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

      3. associate-*l* 39.8%

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

      4. unpow2 39.8%

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

      5. unpow2 39.8%

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

      6. difference-of-squares 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in angle around 0 44.0%

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

   5. Taylor expanded in a around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. *-commutative 45.8%

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

      3. unpow2 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   9. 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* 45.8%

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

      3. unpow2 45.8%

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

      4. *-commutative 45.8%

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

      5. associate-*l* 45.7%

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

      6. *-commutative 45.7%

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

      7. associate-*r* 60.7%

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

      8. *-commutative 60.7%

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

   10. Simplified 60.7%

       \[\leadsto \color{blue}{\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+80}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(-a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 17: 38.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+99}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 2e+99)
   (* 0.011111111111111112 (* PI (* angle (* b b))))
   (* PI (* 0.011111111111111112 (* b (* b angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+99) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
	} else {
		tmp = ((double) M_PI) * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+99) {
		tmp = 0.011111111111111112 * (Math.PI * (angle * (b * b)));
	} else {
		tmp = Math.PI * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 2e+99:
		tmp = 0.011111111111111112 * (math.pi * (angle * (b * b)))
	else:
		tmp = math.pi * (0.011111111111111112 * (b * (b * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 2e+99)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))));
	else
		tmp = Float64(pi * Float64(0.011111111111111112 * Float64(b * Float64(b * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2e+99)
		tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
	else
		tmp = pi * (0.011111111111111112 * (b * (b * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 2e+99], N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.011111111111111112 * N[(b * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999999e99

   1. Initial program 59.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 59.7%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 59.7%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 59.7%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 59.7%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 59.7%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 60.2%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 55.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0 30.1%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 30.1%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \]

      2. *-commutative 30.1%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)} \]

      3. unpow2 30.1%

         \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

   7. Simplified 30.1%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

   if 1.9999999999999999e99 < b

   1. Initial program 39.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 39.9%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 39.9%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 39.9%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 39.9%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 39.9%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 47.9%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 47.9%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 44.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0 46.6%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 46.6%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \]

      2. *-commutative 46.6%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)} \]

      3. unpow2 46.6%

         \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

   7. Simplified 46.6%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 46.6%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 46.6%

         \[\leadsto \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right) \cdot 0.011111111111111112} \]

      2. associate-*r* 46.6%

         \[\leadsto \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \cdot 0.011111111111111112 \]

      3. unpow2 46.6%

         \[\leadsto \left(\left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112 \]

      4. *-commutative 46.6%

         \[\leadsto \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \cdot 0.011111111111111112 \]

      5. associate-*l* 46.6%

         \[\leadsto \color{blue}{\pi \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right)} \]

      6. *-commutative 46.6%

         \[\leadsto \pi \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

      7. associate-*r* 62.7%

         \[\leadsto \pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot b\right)}\right) \]

      8. *-commutative 62.7%

         \[\leadsto \pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot b\right)\right)}\right) \]

   10. Simplified 62.7%

       \[\leadsto \color{blue}{\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+99}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 18: 50.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{+77}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.08e+77)
   (* (* PI angle) (* -0.011111111111111112 (* a a)))
   (* PI (* 0.011111111111111112 (* b (* b angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.08e+77) {
		tmp = (((double) M_PI) * angle) * (-0.011111111111111112 * (a * a));
	} else {
		tmp = ((double) M_PI) * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.08e+77) {
		tmp = (Math.PI * angle) * (-0.011111111111111112 * (a * a));
	} else {
		tmp = Math.PI * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 1.08e+77:
		tmp = (math.pi * angle) * (-0.011111111111111112 * (a * a))
	else:
		tmp = math.pi * (0.011111111111111112 * (b * (b * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.08e+77)
		tmp = Float64(Float64(pi * angle) * Float64(-0.011111111111111112 * Float64(a * a)));
	else
		tmp = Float64(pi * Float64(0.011111111111111112 * Float64(b * Float64(b * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.08e+77)
		tmp = (pi * angle) * (-0.011111111111111112 * (a * a));
	else
		tmp = pi * (0.011111111111111112 * (b * (b * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 1.08e+77], N[(N[(Pi * angle), $MachinePrecision] * N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.011111111111111112 * N[(b * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.07999999999999996e77

   1. Initial program 60.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 60.0%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 60.0%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 60.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 60.0%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 60.0%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 60.5%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 60.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 55.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   5. Taylor expanded in a around inf 41.7%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 41.7%

         \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot {a}^{2}\right) \cdot \left(angle \cdot \pi\right)} \]

      2. unpow2 41.7%

         \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(angle \cdot \pi\right) \]

   7. Simplified 41.7%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \pi\right)} \]

   if 1.07999999999999996e77 < b

   1. Initial program 39.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 39.8%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 39.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 39.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 39.8%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 39.8%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 47.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 44.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0 45.8%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 45.8%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \]

      2. *-commutative 45.8%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)} \]

      3. unpow2 45.8%

         \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

   7. Simplified 45.8%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   9. 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* 45.8%

         \[\leadsto \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \cdot 0.011111111111111112 \]

      3. unpow2 45.8%

         \[\leadsto \left(\left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112 \]

      4. *-commutative 45.8%

         \[\leadsto \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \cdot 0.011111111111111112 \]

      5. associate-*l* 45.7%

         \[\leadsto \color{blue}{\pi \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right)} \]

      6. *-commutative 45.7%

         \[\leadsto \pi \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

      7. associate-*r* 60.7%

         \[\leadsto \pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot b\right)}\right) \]

      8. *-commutative 60.7%

         \[\leadsto \pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot b\right)\right)}\right) \]

   10. Simplified 60.7%

       \[\leadsto \color{blue}{\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{+77}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 19: 50.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+77}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= b 3.7e+77)
   (* -0.011111111111111112 (* PI (* angle (* a a))))
   (* PI (* 0.011111111111111112 (* b (* b angle))))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.7e+77) {
		tmp = -0.011111111111111112 * (((double) M_PI) * (angle * (a * a)));
	} else {
		tmp = ((double) M_PI) * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3.7e+77) {
		tmp = -0.011111111111111112 * (Math.PI * (angle * (a * a)));
	} else {
		tmp = Math.PI * (0.011111111111111112 * (b * (b * angle)));
	}
	return tmp;
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	tmp = 0
	if b <= 3.7e+77:
		tmp = -0.011111111111111112 * (math.pi * (angle * (a * a)))
	else:
		tmp = math.pi * (0.011111111111111112 * (b * (b * angle)))
	return tmp

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	tmp = 0.0
	if (b <= 3.7e+77)
		tmp = Float64(-0.011111111111111112 * Float64(pi * Float64(angle * Float64(a * a))));
	else
		tmp = Float64(pi * Float64(0.011111111111111112 * Float64(b * Float64(b * angle))));
	end
	return tmp
end

MATLAB

a = abs(a)
b = abs(b)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 3.7e+77)
		tmp = -0.011111111111111112 * (pi * (angle * (a * a)));
	else
		tmp = pi * (0.011111111111111112 * (b * (b * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[b, 3.7e+77], N[(-0.011111111111111112 * N[(Pi * N[(angle * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.011111111111111112 * N[(b * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.69999999999999995e77

   1. Initial program 60.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 60.0%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 60.0%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 60.0%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 60.0%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 60.0%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 60.5%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 60.5%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 55.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   5. Taylor expanded in a around inf 41.7%

      \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 41.7%

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112} \]

      2. associate-*r* 41.7%

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot angle\right) \cdot \pi\right)} \cdot -0.011111111111111112 \]

      3. *-commutative 41.7%

         \[\leadsto \color{blue}{\left(\pi \cdot \left({a}^{2} \cdot angle\right)\right)} \cdot -0.011111111111111112 \]

      4. *-commutative 41.7%

         \[\leadsto \left(\pi \cdot \color{blue}{\left(angle \cdot {a}^{2}\right)}\right) \cdot -0.011111111111111112 \]

      5. unpow2 41.7%

         \[\leadsto \left(\pi \cdot \left(angle \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot -0.011111111111111112 \]

   7. Simplified 41.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112} \]

   if 3.69999999999999995e77 < b

   1. Initial program 39.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. *-commutative 39.8%

         \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. associate-*l* 39.8%

         \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. associate-*l* 39.8%

         \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      4. unpow2 39.8%

         \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. unpow2 39.8%

         \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. difference-of-squares 47.3%

         \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0 44.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   5. Taylor expanded in a around 0 45.8%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 45.8%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \]

      2. *-commutative 45.8%

         \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)} \]

      3. unpow2 45.8%

         \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

   7. Simplified 45.8%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
   9. 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* 45.8%

         \[\leadsto \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \cdot 0.011111111111111112 \]

      3. unpow2 45.8%

         \[\leadsto \left(\left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \pi\right) \cdot 0.011111111111111112 \]

      4. *-commutative 45.8%

         \[\leadsto \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \cdot 0.011111111111111112 \]

      5. associate-*l* 45.7%

         \[\leadsto \color{blue}{\pi \cdot \left(\left(angle \cdot \left(b \cdot b\right)\right) \cdot 0.011111111111111112\right)} \]

      6. *-commutative 45.7%

         \[\leadsto \pi \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

      7. associate-*r* 60.7%

         \[\leadsto \pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot b\right) \cdot b\right)}\right) \]

      8. *-commutative 60.7%

         \[\leadsto \pi \cdot \left(0.011111111111111112 \cdot \color{blue}{\left(b \cdot \left(angle \cdot b\right)\right)}\right) \]

   10. Simplified 60.7%

       \[\leadsto \color{blue}{\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+77}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 20: 34.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

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 (* PI (* b b)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * (b * b)));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * (b * b)))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
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[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 56.6%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 56.6%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 56.6%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 56.6%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 56.6%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 58.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 53.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

5. Taylor expanded in a around 0 32.7%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 32.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\left(\pi \cdot {b}^{2}\right)}\right) \]

   2. unpow2 32.7%

      \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

7. Simplified 32.7%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]

8. Final simplification 32.7%

   \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \]

Alternative 21: 34.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} a = |a|\\ b = |b|\\ \\ 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

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 (* angle (* b b)))))

C

a = abs(a);
b = abs(b);
double code(double a, double b, double angle) {
	return 0.011111111111111112 * (((double) M_PI) * (angle * (b * b)));
}

Java

a = Math.abs(a);
b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (Math.PI * (angle * (b * b)));
}

Python

a = abs(a)
b = abs(b)
def code(a, b, angle):
	return 0.011111111111111112 * (math.pi * (angle * (b * b)))

Julia

a = abs(a)
b = abs(b)
function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(pi * Float64(angle * Float64(b * b))))
end

MATLAB

a = abs(a)
b = abs(b)
function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (pi * (angle * (b * b)));
end

Wolfram

NOTE: a should be positive before calling this function
NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(0.011111111111111112 * N[(Pi * N[(angle * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative 56.6%

      \[\leadsto \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. associate-*l* 56.6%

      \[\leadsto \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. associate-*l* 56.6%

      \[\leadsto \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. unpow2 56.6%

      \[\leadsto \left(\color{blue}{b \cdot b} - {a}^{2}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. unpow2 56.6%

      \[\leadsto \left(b \cdot b - \color{blue}{a \cdot a}\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   6. difference-of-squares 58.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in angle around 0 53.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

5. Taylor expanded in a around 0 32.7%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* 32.7%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot {b}^{2}\right) \cdot \pi\right)} \]

   2. *-commutative 32.7%

      \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)} \]

   3. unpow2 32.7%

      \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

7. Simplified 32.7%

   \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)} \]

8. Final simplification 32.7%

   \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \]

Reproduce

herbie shell --seed 2023268 
(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)))))