ab-angle->ABCF A

Average Error: 20.0 → 20.1

Time: 18.9s

Precision: binary64

Cost: 65088

Math

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

↓

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

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* angle (/ PI 180.0)))) 2.0)
  (pow
   (*
    b
    (cos
     (pow (pow (cbrt (cbrt (* angle (* PI 0.005555555555555556)))) 3.0) 3.0)))
   2.0)))

C

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

↓

double code(double a, double b, double angle) {
	return pow((a * sin((angle * (((double) M_PI) / 180.0)))), 2.0) + pow((b * cos(pow(pow(cbrt(cbrt((angle * (((double) M_PI) * 0.005555555555555556)))), 3.0), 3.0))), 2.0);
}

Java

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

↓

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((angle * (Math.PI / 180.0)))), 2.0) + Math.pow((b * Math.cos(Math.pow(Math.pow(Math.cbrt(Math.cbrt((angle * (Math.PI * 0.005555555555555556)))), 3.0), 3.0))), 2.0);
}

Julia

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

↓

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(angle * Float64(pi / 180.0)))) ^ 2.0) + (Float64(b * cos(((cbrt(cbrt(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 3.0) ^ 3.0))) ^ 2.0))
end

Wolfram

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

↓

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[Power[N[Power[N[Power[N[Power[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0

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

2. Applied egg-rr 20.1

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

3. Applied egg-rr 20.1

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

4. Taylor expanded in angle around inf 20.1

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

5. Simplified 20.1

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

   [Start] 20.1	\[ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)}^{3}\right)\right)}^{2} \]
   metadata-eval [<=] 20.1	\[ {\left(a \cdot \sin \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)}^{3}\right)\right)}^{2} \]
   associate-/r/ [<=] 20.1	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)}\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)}^{3}\right)\right)}^{2} \]
   associate-/l* [<=] 20.1	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{1 \cdot \left(angle \cdot \pi\right)}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)}^{3}\right)\right)}^{2} \]
   *-commutative [<=] 20.1	\[ {\left(a \cdot \sin \left(\frac{\color{blue}{\left(angle \cdot \pi\right) \cdot 1}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)}^{3}\right)\right)}^{2} \]
   *-rgt-identity [=>] 20.1	\[ {\left(a \cdot \sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)}^{3}\right)\right)}^{2} \]
   associate-*r/ [<=] 20.1	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left({\left({\left(\sqrt[3]{\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}}\right)}^{3}\right)}^{3}\right)\right)}^{2} \]

6. Final simplification 20.1

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

Alternatives

Alternative 1
Error	20.1
Cost	52224

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

Alternative 2
Error	20.0
Cost	39360

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

Alternative 3
Error	20.1
Cost	26240

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

Alternative 4
Error	20.4
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -1300000 \lor \neg \left(angle \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]

Alternative 5
Error	20.5
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.35 \cdot 10^{+19} \lor \neg \left(angle \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]

Alternative 6
Error	23.4
Cost	20361

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+154} \lor \neg \left(a \leq 10^{+139}\right):\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({\pi}^{2} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	26.0
Cost	20096

\[{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right) \]

Alternative 8
Error	26.0
Cost	19840

\[{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022354 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))