ab-angle->ABCF B

Average Error: 31.1 → 21.6

Time: 21.9s

Precision: binary64

Cost: 111168

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\\ t_1 := e^{t_0}\\ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{-1 + {t_1}^{3}}{1 + \left(t_1 + e^{t_0 \cdot 2}\right)}}\right)}^{3} \end{array} \]

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 (log1p (cos (* angle (* 0.005555555555555556 PI)))))
        (t_1 (exp t_0)))
   (*
    (*
     (* -2.0 (+ b a))
     (* (- a b) (sin (* 0.005555555555555556 (* angle PI)))))
    (pow
     (cbrt (/ (+ -1.0 (pow t_1 3.0)) (+ 1.0 (+ t_1 (exp (* t_0 2.0))))))
     3.0))))

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 = log1p(cos((angle * (0.005555555555555556 * ((double) M_PI)))));
	double t_1 = exp(t_0);
	return ((-2.0 * (b + a)) * ((a - b) * sin((0.005555555555555556 * (angle * ((double) M_PI)))))) * pow(cbrt(((-1.0 + pow(t_1, 3.0)) / (1.0 + (t_1 + exp((t_0 * 2.0)))))), 3.0);
}

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.log1p(Math.cos((angle * (0.005555555555555556 * Math.PI))));
	double t_1 = Math.exp(t_0);
	return ((-2.0 * (b + a)) * ((a - b) * Math.sin((0.005555555555555556 * (angle * Math.PI))))) * Math.pow(Math.cbrt(((-1.0 + Math.pow(t_1, 3.0)) / (1.0 + (t_1 + Math.exp((t_0 * 2.0)))))), 3.0);
}

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 = log1p(cos(Float64(angle * Float64(0.005555555555555556 * pi))))
	t_1 = exp(t_0)
	return Float64(Float64(Float64(-2.0 * Float64(b + a)) * Float64(Float64(a - b) * sin(Float64(0.005555555555555556 * Float64(angle * pi))))) * (cbrt(Float64(Float64(-1.0 + (t_1 ^ 3.0)) / Float64(1.0 + Float64(t_1 + exp(Float64(t_0 * 2.0)))))) ^ 3.0))
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[Log[1 + N[Cos[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, N[(N[(N[(-2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(N[(-1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 + N[Exp[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 31.1

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

2. Simplified 31.1

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

   [Start] 31.1	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.1	\[ \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   sub-neg [=>] 31.1	\[ \left(\left(\color{blue}{\left({b}^{2} + \left(-{a}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   +-commutative [=>] 31.1	\[ \left(\left(\color{blue}{\left(\left(-{a}^{2}\right) + {b}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   neg-sub0 [=>] 31.1	\[ \left(\left(\left(\color{blue}{\left(0 - {a}^{2}\right)} + {b}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-+l- [=>] 31.1	\[ \left(\left(\color{blue}{\left(0 - \left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   sub0-neg [=>] 31.1	\[ \left(\left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   distribute-lft-neg-out [=>] 31.1	\[ \left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   distribute-rgt-neg-in [=>] 31.1	\[ \left(\color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(-2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.1	\[ \left(\left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.1	\[ \left(\left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   metadata-eval [=>] 31.1	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot \color{blue}{-2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Taylor expanded in angle around inf 31.2

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

4. Simplified 21.6

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

   [Start] 31.2	\[ \left(-2 \cdot \left(\left({a}^{2} - {b}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.2	\[ \left(-2 \cdot \left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.2	\[ \left(-2 \cdot \left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   difference-of-squares [=>] 31.2	\[ \left(-2 \cdot \left(\color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*r* [=>] 31.2	\[ \left(-2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [<=] 31.2	\[ \left(-2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [<=] 31.2	\[ \left(-2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*l* [=>] 21.6	\[ \left(-2 \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*l* [<=] 21.6	\[ \color{blue}{\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   +-commutative [=>] 21.6	\[ \left(\left(-2 \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*r* [<=] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Applied egg-rr 21.6

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

6. Applied egg-rr 21.6

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

7. Simplified 21.6

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

   [Start] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{{\left(e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} - 1}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right)}}\right)}^{3} \]
   sub-neg [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\color{blue}{{\left(e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} + \left(-1\right)}}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right)}}\right)}^{3} \]
   metadata-eval [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{{\left(e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} + \color{blue}{-1}}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right)}}\right)}^{3} \]
   +-commutative [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\color{blue}{-1 + {\left(e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}}}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right)}}\right)}^{3} \]
   associate-*r* [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{-1 + {\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)}\right)}^{3}}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right)}}\right)}^{3} \]
   *-commutative [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{-1 + {\left(e^{\mathsf{log1p}\left(\cos \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.005555555555555556\right)\right)}\right)}^{3}}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right)}}\right)}^{3} \]
   associate-*l* [=>] 21.7	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{-1 + {\left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}\right)}^{3}}{e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right)}}\right)}^{3} \]
   +-commutative [=>] 21.7	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{-1 + {\left(e^{\mathsf{log1p}\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}}{\color{blue}{\left(1 + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot 1\right) + e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}}}\right)}^{3} \]

8. Final simplification 21.6

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

Alternatives

Alternative 1
Error	21.6
Cost	20288

\[\frac{\frac{b + a}{\frac{{\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}^{-1}}{2 \cdot \left(b - a\right)}}}{2} \]

Alternative 2
Error	21.6
Cost	13952

\[\frac{\frac{b + a}{\frac{1}{\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)} \cdot \frac{0.5}{b - a}}}{2} \]

Alternative 3
Error	21.8
Cost	13833

\[\begin{array}{l} t_0 := \left(b + a\right) \cdot angle\\ \mathbf{if}\;angle \leq -2 \cdot 10^{-10} \lor \neg \left(angle \leq 7.2 \cdot 10^{-51}\right):\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot t_0 - a \cdot t_0\right)\right)\\ \end{array} \]

Alternative 4
Error	22.4
Cost	13696

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

Alternative 5
Error	22.4
Cost	13696

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

Alternative 6
Error	23.6
Cost	13572

\[\begin{array}{l} t_0 := \left(b + a\right) \cdot angle\\ \mathbf{if}\;angle \leq -1.2 \cdot 10^{+31}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;angle \leq 1.25 \cdot 10^{+25}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot t_0 - a \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(a \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]

Alternative 7
Error	25.1
Cost	7684

\[\begin{array}{l} t_0 := \left(b + a\right) \cdot angle\\ \mathbf{if}\;angle \leq 1.08 \cdot 10^{+25}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot t_0 - a \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(a \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]

Alternative 8
Error	29.6
Cost	7433

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+68} \lor \neg \left(b \leq 1.02 \cdot 10^{+133}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	29.4
Cost	7433

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+103} \lor \neg \left(b \leq 7.5 \cdot 10^{+153}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	29.4
Cost	7433

\[\begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+102} \lor \neg \left(b \leq 3.6 \cdot 10^{+114}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	29.3
Cost	7433

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+118} \lor \neg \left(b \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \end{array} \]

Alternative 12
Error	25.1
Cost	7428

\[\begin{array}{l} \mathbf{if}\;angle \leq 1.18 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{b + a}{\frac{45}{angle \cdot \left(\pi \cdot \left(b - a\right)\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(a \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]

Alternative 13
Error	38.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-64} \lor \neg \left(b \leq 10^{+57}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	38.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+102} \lor \neg \left(b \leq 7.2 \cdot 10^{+83}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	38.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+141} \lor \neg \left(b \leq 7.2 \cdot 10^{+154}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 16
Error	34.2
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{+85} \lor \neg \left(b \leq 1.5 \cdot 10^{-57}\right):\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(a \cdot \pi\right)\right) \cdot -0.011111111111111112\right)\\ \end{array} \]

Alternative 17
Error	43.5
Cost	6912

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

Alternative 18
Error	43.5
Cost	6912

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

Reproduce

herbie shell --seed 2023057 
(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)))))