ab-angle->ABCF B

Average Accuracy: 51.5% → 65.6%

Time: 22.6s

Precision: binary64

Cost: 40328

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := {b}^{2} - {a}^{2}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\left(a + b\right) \cdot \left(-2 \cdot \left(\left(a - b\right) \cdot \sin t_0\right)\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{\frac{2}{\sin \left(2 \cdot t_0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{\frac{1}{a + b}}\\ \end{array} \]

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 (* PI (* angle 0.005555555555555556)))
        (t_1 (- (pow b 2.0) (pow a 2.0))))
   (if (<= t_1 -2e-11)
     (* (+ a b) (* -2.0 (* (- a b) (sin t_0))))
     (if (<= t_1 2e+57)
       (/ (* 2.0 (- (* b b) (* a a))) (/ 2.0 (sin (* 2.0 t_0))))
       (/
        (* -2.0 (* (- a b) (sin (* 0.005555555555555556 (* PI angle)))))
        (/ 1.0 (+ a b)))))))

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 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_1 <= -2e-11) {
		tmp = (a + b) * (-2.0 * ((a - b) * sin(t_0)));
	} else if (t_1 <= 2e+57) {
		tmp = (2.0 * ((b * b) - (a * a))) / (2.0 / sin((2.0 * t_0)));
	} else {
		tmp = (-2.0 * ((a - b) * sin((0.005555555555555556 * (((double) M_PI) * angle))))) / (1.0 / (a + b));
	}
	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 = Math.PI * (angle * 0.005555555555555556);
	double t_1 = Math.pow(b, 2.0) - Math.pow(a, 2.0);
	double tmp;
	if (t_1 <= -2e-11) {
		tmp = (a + b) * (-2.0 * ((a - b) * Math.sin(t_0)));
	} else if (t_1 <= 2e+57) {
		tmp = (2.0 * ((b * b) - (a * a))) / (2.0 / Math.sin((2.0 * t_0)));
	} else {
		tmp = (-2.0 * ((a - b) * Math.sin((0.005555555555555556 * (Math.PI * angle))))) / (1.0 / (a + b));
	}
	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 = math.pi * (angle * 0.005555555555555556)
	t_1 = math.pow(b, 2.0) - math.pow(a, 2.0)
	tmp = 0
	if t_1 <= -2e-11:
		tmp = (a + b) * (-2.0 * ((a - b) * math.sin(t_0)))
	elif t_1 <= 2e+57:
		tmp = (2.0 * ((b * b) - (a * a))) / (2.0 / math.sin((2.0 * t_0)))
	else:
		tmp = (-2.0 * ((a - b) * math.sin((0.005555555555555556 * (math.pi * angle))))) / (1.0 / (a + b))
	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(pi * Float64(angle * 0.005555555555555556))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_1 <= -2e-11)
		tmp = Float64(Float64(a + b) * Float64(-2.0 * Float64(Float64(a - b) * sin(t_0))));
	elseif (t_1 <= 2e+57)
		tmp = Float64(Float64(2.0 * Float64(Float64(b * b) - Float64(a * a))) / Float64(2.0 / sin(Float64(2.0 * t_0))));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64(a - b) * sin(Float64(0.005555555555555556 * Float64(pi * angle))))) / Float64(1.0 / Float64(a + b)));
	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 = pi * (angle * 0.005555555555555556);
	t_1 = (b ^ 2.0) - (a ^ 2.0);
	tmp = 0.0;
	if (t_1 <= -2e-11)
		tmp = (a + b) * (-2.0 * ((a - b) * sin(t_0)));
	elseif (t_1 <= 2e+57)
		tmp = (2.0 * ((b * b) - (a * a))) / (2.0 / sin((2.0 * t_0)));
	else
		tmp = (-2.0 * ((a - b) * sin((0.005555555555555556 * (pi * angle))))) / (1.0 / (a + b));
	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[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-11], N[(N[(a + b), $MachinePrecision] * N[(-2.0 * N[(N[(a - b), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+57], N[(N[(2.0 * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[Sin[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(a - b), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -1.99999999999999988e-11

   1. Initial program 42.6%

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

   2. Simplified 42.6%

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

      [Start] 42.6	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 42.6	\[ \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      sub-neg [=>] 42.6	\[ \left(\left(\color{blue}{\left({b}^{2} + \left(-{a}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      +-commutative [=>] 42.6	\[ \left(\left(\color{blue}{\left(\left(-{a}^{2}\right) + {b}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      neg-sub0 [=>] 42.6	\[ \left(\left(\left(\color{blue}{\left(0 - {a}^{2}\right)} + {b}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-+l- [=>] 42.6	\[ \left(\left(\color{blue}{\left(0 - \left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      sub0-neg [=>] 42.6	\[ \left(\left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      distribute-lft-neg-out [=>] 42.6	\[ \left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      distribute-rgt-neg-in [=>] 42.6	\[ \left(\color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(-2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      unpow2 [=>] 42.6	\[ \left(\left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      unpow2 [=>] 42.6	\[ \left(\left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      metadata-eval [=>] 42.6	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot \color{blue}{-2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied egg-rr 19.7%

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

      [Start] 42.6	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot -2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      add-exp-log [=>] 19.5	\[ \color{blue}{e^{\log \left(\left(\left(a \cdot a - b \cdot b\right) \cdot -2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 19.5	\[ e^{\log \left(\color{blue}{\left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      difference-of-squares [=>] 19.5	\[ e^{\log \left(\left(-2 \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*r* [=>] 19.5	\[ e^{\log \left(\color{blue}{\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      div-inv [=>] 19.7	\[ e^{\log \left(\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      metadata-eval [=>] 19.7	\[ e^{\log \left(\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Taylor expanded in angle around 0 19.0%

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

   5. Applied egg-rr 64.6%

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

      [Start] 19.0	\[ e^{\log \left(\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1 \]
      add-exp-log [<=] 41.2	\[ \color{blue}{\left(\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1 \]
      associate-*l* [=>] 64.6	\[ \color{blue}{\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot 1 \]
      *-commutative [=>] 64.6	\[ \left(\color{blue}{\left(\left(a + b\right) \cdot -2\right)} \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 1 \]
      associate-*l* [=>] 64.6	\[ \color{blue}{\left(\left(a + b\right) \cdot \left(-2 \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \cdot 1 \]

   if -1.99999999999999988e-11 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 2.0000000000000001e57

   1. Initial program 66.8%

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

   2. Simplified 66.8%

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

      [Start] 66.8	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 66.8	\[ \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      unpow2 [=>] 66.8	\[ \left(2 \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
      unpow2 [=>] 66.8	\[ \left(2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   3. Applied egg-rr 66.6%

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

      [Start] 66.8	\[ \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
      *-commutative [=>] 66.8	\[ \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)} \]
      sin-cos-mult [=>] 66.9	\[ \color{blue}{\frac{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}{2}} \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \]
      clear-num [=>] 66.8	\[ \color{blue}{\frac{1}{\frac{2}{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}}} \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \]
      associate-*l/ [=>] 66.8	\[ \color{blue}{\frac{1 \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}{\frac{2}{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}}} \]
      *-un-lft-identity [<=] 66.8	\[ \frac{\color{blue}{2 \cdot \left(b \cdot b - a \cdot a\right)}}{\frac{2}{\sin \left(\pi \cdot \frac{angle}{180} - \pi \cdot \frac{angle}{180}\right) + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}} \]
      +-inverses [=>] 66.8	\[ \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{\frac{2}{\sin \color{blue}{0} + \sin \left(\pi \cdot \frac{angle}{180} + \pi \cdot \frac{angle}{180}\right)}} \]
      count-2 [=>] 66.8	\[ \frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{\frac{2}{\sin 0 + \sin \color{blue}{\left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)}}} \]

   if 2.0000000000000001e57 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 38.7%

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

   2. Simplified 38.7%

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

      [Start] 38.7	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 38.7	\[ \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      sub-neg [=>] 38.7	\[ \left(\left(\color{blue}{\left({b}^{2} + \left(-{a}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      +-commutative [=>] 38.7	\[ \left(\left(\color{blue}{\left(\left(-{a}^{2}\right) + {b}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      neg-sub0 [=>] 38.7	\[ \left(\left(\left(\color{blue}{\left(0 - {a}^{2}\right)} + {b}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-+l- [=>] 38.7	\[ \left(\left(\color{blue}{\left(0 - \left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      sub0-neg [=>] 38.7	\[ \left(\left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      distribute-lft-neg-out [=>] 38.7	\[ \left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      distribute-rgt-neg-in [=>] 38.7	\[ \left(\color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(-2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      unpow2 [=>] 38.7	\[ \left(\left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      unpow2 [=>] 38.7	\[ \left(\left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      metadata-eval [=>] 38.7	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot \color{blue}{-2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied egg-rr 33.1%

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

      [Start] 38.7	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot -2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 38.7	\[ \color{blue}{\left(\left(a \cdot a - b \cdot b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      difference-of-squares [=>] 38.7	\[ \left(\color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)} \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 66.8	\[ \color{blue}{\left(\left(a + b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      flip-+ [=>] 38.7	\[ \left(\color{blue}{\frac{a \cdot a - b \cdot b}{a - b}} \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l/ [=>] 33.0	\[ \color{blue}{\frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}{a - b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      div-inv [=>] 33.1	\[ \frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      metadata-eval [=>] 33.1	\[ \frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. Simplified 66.7%

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

      [Start] 33.1	\[ \frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 33.1	\[ \frac{\color{blue}{\left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(a \cdot a - b \cdot b\right)}}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-/l* [=>] 38.7	\[ \color{blue}{\frac{\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}{\frac{a - b}{a \cdot a - b \cdot b}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 38.7	\[ \frac{\color{blue}{\left(-2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a - b\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*l* [=>] 38.7	\[ \frac{\color{blue}{-2 \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a - b\right)\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [<=] 38.7	\[ \frac{-2 \cdot \color{blue}{\left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 38.7	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 38.7	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-*r* [<=] 38.5	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      difference-of-squares [=>] 38.5	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-commutative [=>] 38.5	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{a - b}{\color{blue}{\left(a - b\right) \cdot \left(a + b\right)}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      associate-/r* [=>] 66.7	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\color{blue}{\frac{\frac{a - b}{a - b}}{a + b}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      *-inverses [=>] 66.7	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{\color{blue}{1}}{a + b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      +-commutative [=>] 66.7	\[ \frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}{\frac{1}{\color{blue}{b + a}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. Taylor expanded in angle around 0 65.3%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\left(a + b\right) \cdot \left(-2 \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;\frac{2 \cdot \left(b \cdot b - a \cdot a\right)}{\frac{2}{\sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}{\frac{1}{a + b}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	66.3%
Cost	27072

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

Alternative 2
Accuracy	66.6%
Cost	26816

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

Alternative 3
Accuracy	66.6%
Cost	26816

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

Alternative 4
Accuracy	66.2%
Cost	14089

\[\begin{array}{l} t_0 := angle \cdot \left(a + b\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-47} \lor \neg \left(\frac{angle}{180} \leq 10^{-17}\right):\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot t_0 - a \cdot t_0\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	65.2%
Cost	13696

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

Alternative 6
Accuracy	64.3%
Cost	13508

\[\begin{array}{l} t_0 := angle \cdot \left(a + b\right)\\ \mathbf{if}\;angle \leq -0.1:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(-\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 450:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot t_0 - a \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-a\right)\\ \end{array} \]

Alternative 7
Accuracy	64.2%
Cost	7817

\[\begin{array}{l} t_0 := angle \cdot \left(a + b\right)\\ \mathbf{if}\;angle \leq -980 \lor \neg \left(angle \leq 450\right):\\ \;\;\;\;a \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot t_0 - a \cdot t_0\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	54.1%
Cost	7432

\[\begin{array}{l} t_0 := a \cdot \left(angle \cdot -0.011111111111111112\right)\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;\pi \cdot \left(a \cdot t_0\right)\\ \mathbf{elif}\;a \leq 1.22 \cdot 10^{+101}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\pi \cdot t_0\right)\\ \end{array} \]

Alternative 9
Accuracy	54.2%
Cost	7432

\[\begin{array}{l} t_0 := a \cdot \left(angle \cdot -0.011111111111111112\right)\\ \mathbf{if}\;a \leq -1.32 \cdot 10^{+45}:\\ \;\;\;\;\pi \cdot \left(a \cdot t_0\right)\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{+101}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\pi \cdot t_0\right)\\ \end{array} \]

Alternative 10
Accuracy	54.2%
Cost	7432

\[\begin{array}{l} t_0 := a \cdot \left(angle \cdot -0.011111111111111112\right)\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{+44}:\\ \;\;\;\;\pi \cdot \left(a \cdot t_0\right)\\ \mathbf{elif}\;a \leq 1.38 \cdot 10^{+101}:\\ \;\;\;\;angle \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\pi \cdot t_0\right)\\ \end{array} \]

Alternative 11
Accuracy	40.9%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.15 \cdot 10^{-45} \lor \neg \left(angle \leq 1.35 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	49.4%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1.95 \cdot 10^{-52} \lor \neg \left(a \leq 5.8 \cdot 10^{-18}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	49.4%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-52} \lor \neg \left(a \leq 4.1 \cdot 10^{-18}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	49.4%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-52} \lor \neg \left(a \leq 5 \cdot 10^{-18}\right):\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	49.5%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-18}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\pi \cdot \left(a \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	19.5%
Cost	448

\[0 \cdot \left(-2 \cdot \left(a \cdot a\right)\right) \]

Alternative 17
Accuracy	16.3%
Cost	256

\[a \cdot \left(-a\right) \]

Reproduce

herbie shell --seed 2023126 
(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)))))