ab-angle->ABCF A

Average Error: 20.1 → 20.1

Time: 19.3s

Precision: binary64

Cost: 78080

Math

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

↓

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

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

Derivation

1. Initial program 20.1

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

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

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{\frac{angle}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}}}{1} \cdot \sqrt[3]{\pi \cdot 0.005555555555555556}\right)}\right)}^{2} \]

4. Simplified 20.1

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

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

5. Applied egg-rr 20.1

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

6. Final simplification 20.1

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

Alternatives

Alternative 1
Error	20.1
Cost	39360

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

Alternative 2
Error	20.1
Cost	39360

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

Alternative 3
Error	20.2
Cost	26240

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

Alternative 4
Error	20.2
Cost	26240

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

Alternative 5
Error	20.4
Cost	20488

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.21:\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.05 \cdot 10^{+20}:\\ \;\;\;\;{b}^{2} + \left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	20.4
Cost	20488

\[\begin{array}{l} t_0 := a \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;angle \leq -0.21:\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.05 \cdot 10^{+20}:\\ \;\;\;\;{b}^{2} + \left(t_0 \cdot t_0\right) \cdot {\pi}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	20.5
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.21 \lor \neg \left(angle \leq 1.05 \cdot 10^{+20}\right):\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 8
Error	20.5
Cost	20424

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.21:\\ \;\;\;\;{b}^{2} + a \cdot \left(\left(a \cdot 0.5\right) \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.05 \cdot 10^{+20}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	21.9
Cost	20105

\[\begin{array}{l} \mathbf{if}\;angle \leq -5.2 \cdot 10^{+37} \lor \neg \left(angle \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;{b}^{2} + {\left(a \cdot 0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 10
Error	31.8
Cost	13248

\[{b}^{2} + {\left(a \cdot 0\right)}^{2} \]

Reproduce

herbie shell --seed 2023066 
(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)))