ab-angle->ABCF B

Average Accuracy: 51.4% → 65.9%

Time: 22.5s

Precision: binary64

Cost: 20228

Math

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

↓

\[\begin{array}{l} t_0 := b \cdot b - a \cdot a\\ \mathbf{if}\;\frac{angle}{180} \leq -0.0005:\\ \;\;\;\;\left|t_0 \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-47}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot 2}{\frac{2}{\sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (* b b) (* a a))))
   (if (<= (/ angle 180.0) -0.0005)
     (fabs (* t_0 (sin (* (* PI angle) 0.011111111111111112))))
     (if (<= (/ angle 180.0) 1e-47)
       (* -0.011111111111111112 (* (- a b) (* angle (* PI (+ a b)))))
       (/
        (* t_0 2.0)
        (/ 2.0 (sin (* (* PI (* angle 0.005555555555555556)) 2.0))))))))

C

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

↓

double code(double a, double b, double angle) {
	double t_0 = (b * b) - (a * a);
	double tmp;
	if ((angle / 180.0) <= -0.0005) {
		tmp = fabs((t_0 * sin(((((double) M_PI) * angle) * 0.011111111111111112))));
	} else if ((angle / 180.0) <= 1e-47) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (((double) M_PI) * (a + b))));
	} else {
		tmp = (t_0 * 2.0) / (2.0 / sin(((((double) M_PI) * (angle * 0.005555555555555556)) * 2.0)));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle) {
	double t_0 = (b * b) - (a * a);
	double tmp;
	if ((angle / 180.0) <= -0.0005) {
		tmp = Math.abs((t_0 * Math.sin(((Math.PI * angle) * 0.011111111111111112))));
	} else if ((angle / 180.0) <= 1e-47) {
		tmp = -0.011111111111111112 * ((a - b) * (angle * (Math.PI * (a + b))));
	} else {
		tmp = (t_0 * 2.0) / (2.0 / Math.sin(((Math.PI * (angle * 0.005555555555555556)) * 2.0)));
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle):
	t_0 = (b * b) - (a * a)
	tmp = 0
	if (angle / 180.0) <= -0.0005:
		tmp = math.fabs((t_0 * math.sin(((math.pi * angle) * 0.011111111111111112))))
	elif (angle / 180.0) <= 1e-47:
		tmp = -0.011111111111111112 * ((a - b) * (angle * (math.pi * (a + b))))
	else:
		tmp = (t_0 * 2.0) / (2.0 / math.sin(((math.pi * (angle * 0.005555555555555556)) * 2.0)))
	return tmp

Julia

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

↓

function code(a, b, angle)
	t_0 = Float64(Float64(b * b) - Float64(a * a))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -0.0005)
		tmp = abs(Float64(t_0 * sin(Float64(Float64(pi * angle) * 0.011111111111111112))));
	elseif (Float64(angle / 180.0) <= 1e-47)
		tmp = Float64(-0.011111111111111112 * Float64(Float64(a - b) * Float64(angle * Float64(pi * Float64(a + b)))));
	else
		tmp = Float64(Float64(t_0 * 2.0) / Float64(2.0 / sin(Float64(Float64(pi * Float64(angle * 0.005555555555555556)) * 2.0))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle)
	t_0 = (b * b) - (a * a);
	tmp = 0.0;
	if ((angle / 180.0) <= -0.0005)
		tmp = abs((t_0 * sin(((pi * angle) * 0.011111111111111112))));
	elseif ((angle / 180.0) <= 1e-47)
		tmp = -0.011111111111111112 * ((a - b) * (angle * (pi * (a + b))));
	else
		tmp = (t_0 * 2.0) / (2.0 / sin(((pi * (angle * 0.005555555555555556)) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -0.0005], N[Abs[N[(t$95$0 * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-47], N[(-0.011111111111111112 * N[(N[(a - b), $MachinePrecision] * N[(angle * N[(Pi * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 2.0), $MachinePrecision] / N[(2.0 / N[Sin[N[(N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -5.0000000000000001e-4

   1. Initial program 22.8%

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

   2. Simplified 22.8%

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

      [Start] 22.8	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 22.8	\[ \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      unpow2 [=>] 22.8	\[ \left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
      unpow2 [=>] 22.8	\[ \left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\left(\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.5\right)\right)}^{2}}} \]
      Proof

      [Start] 22.8	\[ \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
      add-sqr-sqrt [=>] 15.6	\[ \color{blue}{\sqrt{\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \sqrt{\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}} \]
      sqrt-unprod [=>] 15.8	\[ \color{blue}{\sqrt{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}} \]
      pow2 [=>] 15.8	\[ \sqrt{\color{blue}{{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}^{2}}} \]

   4. Simplified 20.2%

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

      [Start] 15.8	\[ \sqrt{{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\left(\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.5\right)\right)}^{2}} \]
      unpow2 [=>] 15.8	\[ \sqrt{\color{blue}{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\left(\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.5\right)\right) \cdot \left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\left(\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.5\right)\right)}} \]
      rem-sqrt-square [=>] 20.3	\[ \color{blue}{\left|\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\left(\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.5\right)\right|} \]
      *-commutative [=>] 20.3	\[ \left|\color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot 2\right)} \cdot \left(\left(\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.5\right)\right| \]
      associate-*l* [=>] 20.3	\[ \left|\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot \left(2 \cdot \left(\left(\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.5\right)\right)}\right| \]

   if -5.0000000000000001e-4 < (/.f64 angle 180) < 9.9999999999999997e-48

   1. Initial program 71.2%

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

   2. Simplified 71.2%

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

      [Start] 71.2	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 71.2	\[ \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      sub-neg [=>] 71.2	\[ \left(\left(\color{blue}{\left({b}^{2} + \left(-{a}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      +-commutative [=>] 71.2	\[ \left(\left(\color{blue}{\left(\left(-{a}^{2}\right) + {b}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      neg-sub0 [=>] 71.2	\[ \left(\left(\left(\color{blue}{\left(0 - {a}^{2}\right)} + {b}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-+l- [=>] 71.2	\[ \left(\left(\color{blue}{\left(0 - \left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      sub0-neg [=>] 71.2	\[ \left(\left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      distribute-lft-neg-out [=>] 71.2	\[ \left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      distribute-rgt-neg-in [=>] 71.2	\[ \left(\color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(-2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      unpow2 [=>] 71.2	\[ \left(\left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      unpow2 [=>] 71.2	\[ \left(\left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      metadata-eval [=>] 71.2	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot \color{blue}{-2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied egg-rr 64.1%

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

      [Start] 71.2	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot -2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 71.2	\[ \color{blue}{\left(\left(a \cdot a - b \cdot b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      difference-of-squares [=>] 71.2	\[ \left(\color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)} \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 99.4	\[ \color{blue}{\left(\left(a + b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      flip-+ [=>] 71.2	\[ \left(\color{blue}{\frac{a \cdot a - b \cdot b}{a - b}} \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l/ [=>] 64.1	\[ \color{blue}{\frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}{a - b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      div-inv [=>] 64.1	\[ \frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      metadata-eval [=>] 64.1	\[ \frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 99.4%

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

      [Start] 64.1	\[ \frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 64.1	\[ \frac{\color{blue}{\left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot a - b \cdot b\right)}}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-/l* [=>] 71.2	\[ \color{blue}{\frac{\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}{\frac{a - b}{a \cdot a - b \cdot b}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 71.2	\[ \frac{\color{blue}{\left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a - b\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 71.2	\[ \frac{\color{blue}{-2 \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a - b\right)\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [<=] 71.2	\[ \frac{-2 \cdot \color{blue}{\left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 71.2	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 71.2	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*r* [<=] 71.2	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      difference-of-squares [=>] 71.2	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 71.2	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{a - b}{\color{blue}{\left(a - b\right) \cdot \left(a + b\right)}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-/r* [=>] 99.4	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\frac{\frac{a - b}{a - b}}{a + b}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-inverses [=>] 99.4	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{\color{blue}{1}}{a + b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      +-commutative [=>] 99.4	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{1}{\color{blue}{b + a}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0 71.1%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. Simplified 99.1%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot angle\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      Proof

      [Start] 71.1	\[ \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 71.1	\[ \left(-0.011111111111111112 \cdot \color{blue}{\left(\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right) \cdot angle\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 99.1	\[ \left(-0.011111111111111112 \cdot \color{blue}{\left(\left(a - b\right) \cdot \left(\left(\left(a + b\right) \cdot \pi\right) \cdot angle\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 99.1	\[ \left(-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\color{blue}{\left(\pi \cdot \left(a + b\right)\right)} \cdot angle\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      +-commutative [=>] 99.1	\[ \left(-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(\left(\pi \cdot \color{blue}{\left(b + a\right)}\right) \cdot angle\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Taylor expanded in angle around 0 99.1%

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

   if 9.9999999999999997e-48 < (/.f64 angle 180)

   1. Initial program 31.8%

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

   2. Simplified 31.8%

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

      [Start] 31.8	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 31.8	\[ \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      unpow2 [=>] 31.8	\[ \left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
      unpow2 [=>] 31.8	\[ \left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{\frac{2}{\sin 0 + \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}} \]
      Proof

      [Start] 31.8	\[ \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
      *-commutative [=>] 31.8	\[ \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)} \]
      sin-cos-mult [=>] 31.8	\[ \color{blue}{\frac{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}{2}} \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \]
      clear-num [=>] 31.7	\[ \color{blue}{\frac{1}{\frac{2}{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}}} \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \]
      associate-*l/ [=>] 31.7	\[ \color{blue}{\frac{1 \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}{\frac{2}{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}}} \]
      *-un-lft-identity [<=] 31.7	\[ \frac{\color{blue}{2 \cdot \left(b \cdot b - a \cdot a\right)}}{\frac{2}{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}} \]
      +-inverses [=>] 31.7	\[ \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{\frac{2}{\sin \color{blue}{0} + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}} \]
      count-2 [=>] 31.7	\[ \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{\frac{2}{\sin 0 + \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -0.0005:\\ \;\;\;\;\left|\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right|\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-47}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot b - a \cdot a\right) \cdot 2}{\frac{2}{\sin \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	66.7%
Cost	40200

\[\begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ t_1 := \frac{1}{a + b}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{t_1}\\ \mathbf{elif}\;t_0 \leq 10^{+296}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}{t_1}\\ \end{array} \]

Alternative 2
Accuracy	66.7%
Cost	26944

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin t_0\right)}{\frac{1}{a + b}} \cdot \cos t_0 \end{array} \]

Alternative 3
Accuracy	66.7%
Cost	26816

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

Alternative 4
Accuracy	66.7%
Cost	26816

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

Alternative 5
Accuracy	66.4%
Cost	14089

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-29} \lor \neg \left(\frac{angle}{180} \leq 10^{-47}\right):\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	64.0%
Cost	13833

\[\begin{array}{l} \mathbf{if}\;angle \leq -34000000 \lor \neg \left(angle \leq 3.9 \cdot 10^{-16}\right):\\ \;\;\;\;2 \cdot \left(\frac{b}{\frac{2}{b}} \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	65.0%
Cost	13824

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

Alternative 8
Accuracy	63.3%
Cost	13705

\[\begin{array}{l} \mathbf{if}\;angle \leq -6.2 \cdot 10^{+38} \lor \neg \left(angle \leq 3.9 \cdot 10^{-16}\right):\\ \;\;\;\;2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	62.3%
Cost	13444

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\left|\pi \cdot \left(\left(b \cdot b\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right|\\ \mathbf{elif}\;angle \leq 1.45 \cdot 10^{+58}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	52.5%
Cost	7564

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+49}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-83}:\\ \;\;\;\;\left(a \cdot \left(a \cdot angle\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+108}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	52.6%
Cost	7564

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+49}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-84}:\\ \;\;\;\;\left(a \cdot \left(a \cdot angle\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+151}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	50.0%
Cost	7440

\[\begin{array}{l} t_0 := \left(a \cdot \left(a \cdot angle\right)\right) \cdot \left(\pi \cdot -0.011111111111111112\right)\\ t_1 := 0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	62.2%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;angle \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(a - b\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	39.6%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+144} \lor \neg \left(b \leq 1.3 \cdot 10^{-185}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	39.9%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+144}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-15}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	39.9%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+135}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+42}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 17
Accuracy	39.9%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+135}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-14}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 18
Accuracy	32.1%
Cost	6912

\[0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \]

Reproduce

herbie shell --seed 2023133 
(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)))))