ab-angle->ABCF A

Average Error: 20.2 → 20.1

Time: 18.0s

Precision: binary64

Cost: 52224

Math

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

↓

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

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 (/ 180.0 PI)))) 2.0)
  (pow
   (* b (cbrt (pow (cos (* PI (* angle 0.005555555555555556))) 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 / (180.0 / ((double) M_PI))))), 2.0) + pow((b * cbrt(pow(cos((((double) M_PI) * (angle * 0.005555555555555556))), 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 / (180.0 / Math.PI)))), 2.0) + Math.pow((b * Math.cbrt(Math.pow(Math.cos((Math.PI * (angle * 0.005555555555555556))), 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(180.0 / pi)))) ^ 2.0) + (Float64(b * cbrt((cos(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 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[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Power[N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.2

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

2. Simplified 20.1

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

   [Start] 20.2	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 20.1	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 20.1	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} \]

3. Applied egg-rr 20.1

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

4. Taylor expanded in angle around inf 20.1

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

5. Simplified 20.1

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

   [Start] 20.1	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{3}}\right)}^{2} \]
   associate-*r* [=>] 20.1	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{3}}\right)}^{2} \]

6. Final simplification 20.1

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

Alternatives

Alternative 1
Error	20.1
Cost	52160

\[{\left(a \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} \]

Alternative 2
Error	20.1
Cost	39360

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

Alternative 3
Error	20.3
Cost	26240

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

Alternative 4
Error	20.3
Cost	20745

\[\begin{array}{l} t_0 := a \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -0.0002 \lor \neg \left(\frac{angle}{180} \leq 10^{-5}\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(t_0 \cdot t_0\right) \cdot {\pi}^{2}\\ \end{array} \]

Alternative 5
Error	20.4
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0048 \lor \neg \left(angle \leq 0.0047\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left({\pi}^{2} \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 6
Error	25.9
Cost	20096

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

Alternative 7
Error	25.9
Cost	19840

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

Alternative 8
Error	25.8
Cost	19840

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

Reproduce

herbie shell --seed 2023073 
(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)))