ab-angle->ABCF C

Average Error: 31.97% → 31.97%

Time: 20.6s

Precision: binary64

Cost: 58880

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{180}{angle}}\\ {\left(a \cdot \cos \left(\frac{\frac{\pi}{{t_0}^{2}}}{t_0}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (/ 180.0 angle))))
   (+
    (pow (* a (cos (/ (/ PI (pow t_0 2.0)) t_0))) 2.0)
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0))))

C

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

↓

double code(double a, double b, double angle) {
	double t_0 = cbrt((180.0 / angle));
	return pow((a * cos(((((double) M_PI) / pow(t_0, 2.0)) / t_0))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

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

↓

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt((180.0 / angle));
	return Math.pow((a * Math.cos(((Math.PI / Math.pow(t_0, 2.0)) / t_0))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Julia

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

↓

function code(a, b, angle)
	t_0 = cbrt(Float64(180.0 / angle))
	return Float64((Float64(a * cos(Float64(Float64(pi / (t_0 ^ 2.0)) / t_0))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

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

↓

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[(180.0 / angle), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(N[(Pi / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 31.97

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

2. Applied egg-rr 31.97

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

3. Applied egg-rr 66.25

   \[\leadsto {\left(a \cdot \cos \left(\frac{\frac{\pi}{\color{blue}{{\left(\frac{180}{angle}\right)}^{0.6666666666666666}}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Simplified 31.97

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

   [Start] 66.25	\[ {\left(a \cdot \cos \left(\frac{\frac{\pi}{{\left(\frac{180}{angle}\right)}^{0.6666666666666666}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   metadata-eval [<=] 66.25	\[ {\left(a \cdot \cos \left(\frac{\frac{\pi}{{\left(\frac{180}{angle}\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   pow-sqr [<=] 66.25	\[ {\left(a \cdot \cos \left(\frac{\frac{\pi}{\color{blue}{{\left(\frac{180}{angle}\right)}^{0.3333333333333333} \cdot {\left(\frac{180}{angle}\right)}^{0.3333333333333333}}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   unpow1/3 [=>] 66.23	\[ {\left(a \cdot \cos \left(\frac{\frac{\pi}{\color{blue}{\sqrt[3]{\frac{180}{angle}}} \cdot {\left(\frac{180}{angle}\right)}^{0.3333333333333333}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   unpow1/3 [=>] 31.97	\[ {\left(a \cdot \cos \left(\frac{\frac{\pi}{\sqrt[3]{\frac{180}{angle}} \cdot \color{blue}{\sqrt[3]{\frac{180}{angle}}}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   unpow2 [<=] 31.97	\[ {\left(a \cdot \cos \left(\frac{\frac{\pi}{\color{blue}{{\left(\sqrt[3]{\frac{180}{angle}}\right)}^{2}}}}{\sqrt[3]{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

5. Final simplification 31.97

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

Alternatives

Alternative 1
Error	31.97%
Cost	58752

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

Alternative 2
Error	31.95%
Cost	39488

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

Alternative 3
Error	31.95%
Cost	39360

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

Alternative 4
Error	31.96%
Cost	39360

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

Alternative 5
Error	31.96%
Cost	39360

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

Alternative 6
Error	31.94%
Cost	26368

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

Alternative 7
Error	31.92%
Cost	26240

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

Alternative 8
Error	31.96%
Cost	26240

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

Alternative 9
Error	31.98%
Cost	20488

\[\begin{array}{l} t_0 := b \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;angle \leq -0.185:\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 0.0056:\\ \;\;\;\;{a}^{2} + t_0 \cdot \left(0.005555555555555556 \cdot \left(0.005555555555555556 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \frac{b \cdot b}{2} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	32.05%
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.185 \lor \neg \left(angle \leq 780\right):\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 11
Error	31.99%
Cost	20424

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.185:\\ \;\;\;\;{a}^{2} + \frac{b}{\frac{2}{b}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 0.0046:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \frac{b \cdot b}{2} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	40.82%
Cost	19840

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

Alternative 13
Error	40.78%
Cost	19840

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

Alternative 14
Error	40.69%
Cost	19840

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

Alternative 15
Error	40.65%
Cost	19840

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

Reproduce

herbie shell --seed 2023121 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))