ab-angle->ABCF A

Average Error: 21.0 → 21.0

Time: 20.0s

Precision: binary64

Cost: 78272

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{1}{\pi}}\\ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{t_0 \cdot \left(t_0 \cdot \sqrt[3]{32400}\right)}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \end{array} \]

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
 (let* ((t_0 (cbrt (/ 1.0 PI))))
   (+
    (pow (* a (sin (/ angle (/ 180.0 PI)))) 2.0)
    (pow
     (*
      b
      (cos (/ (/ angle (* t_0 (* t_0 (cbrt 32400.0)))) (cbrt (/ 180.0 PI)))))
     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) {
	double t_0 = cbrt((1.0 / ((double) M_PI)));
	return pow((a * sin((angle / (180.0 / ((double) M_PI))))), 2.0) + pow((b * cos(((angle / (t_0 * (t_0 * cbrt(32400.0)))) / cbrt((180.0 / ((double) M_PI)))))), 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) {
	double t_0 = Math.cbrt((1.0 / Math.PI));
	return Math.pow((a * Math.sin((angle / (180.0 / Math.PI)))), 2.0) + Math.pow((b * Math.cos(((angle / (t_0 * (t_0 * Math.cbrt(32400.0)))) / Math.cbrt((180.0 / Math.PI))))), 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)
	t_0 = cbrt(Float64(1.0 / pi))
	return Float64((Float64(a * sin(Float64(angle / Float64(180.0 / pi)))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / Float64(t_0 * Float64(t_0 * cbrt(32400.0)))) / cbrt(Float64(180.0 / pi))))) ^ 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_] := Block[{t$95$0 = N[Power[N[(1.0 / Pi), $MachinePrecision], 1/3], $MachinePrecision]}, 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[(t$95$0 * N[(t$95$0 * N[Power[32400.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(180.0 / Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 21.0

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

2. Simplified 21.0

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

   [Start] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 21.0	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} \]

3. Applied egg-rr 21.0

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

4. Simplified 21.0

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

   [Start] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{1}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}} \cdot \frac{angle}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   associate-*r/ [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{\frac{1}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}} \cdot angle}{\sqrt[3]{\frac{180}{\pi}}}\right)}\right)}^{2} \]
   associate-*l/ [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\frac{1 \cdot angle}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   *-lft-identity [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{\color{blue}{angle}}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   /-rgt-identity [<=] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\frac{\frac{angle}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}}}{1}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   /-rgt-identity [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\frac{angle}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]

5. Applied egg-rr 21.0

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\color{blue}{\sqrt[3]{32400} \cdot \left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{\frac{1}{\pi}}\right)}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]

6. Simplified 21.0

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\color{blue}{\sqrt[3]{\frac{1}{\pi}} \cdot \left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{32400}\right)}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   Proof

   [Start] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\sqrt[3]{32400} \cdot \left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{\frac{1}{\pi}}\right)}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   *-commutative [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\color{blue}{\left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{\frac{1}{\pi}}\right) \cdot \sqrt[3]{32400}}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   unpow1/3 [<=] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\frac{1}{\pi}}\right) \cdot \sqrt[3]{32400}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   unpow1/3 [<=] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\left({\left(\frac{1}{\pi}\right)}^{0.3333333333333333} \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{32400}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   associate-*l* [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.3333333333333333} \cdot \left({\left(\frac{1}{\pi}\right)}^{0.3333333333333333} \cdot \sqrt[3]{32400}\right)}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   unpow1/3 [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\color{blue}{\sqrt[3]{\frac{1}{\pi}}} \cdot \left({\left(\frac{1}{\pi}\right)}^{0.3333333333333333} \cdot \sqrt[3]{32400}\right)}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   unpow1/3 [=>] 21.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\sqrt[3]{\frac{1}{\pi}} \cdot \left(\color{blue}{\sqrt[3]{\frac{1}{\pi}}} \cdot \sqrt[3]{32400}\right)}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]

7. Final simplification 21.0

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{\sqrt[3]{\frac{1}{\pi}} \cdot \left(\sqrt[3]{\frac{1}{\pi}} \cdot \sqrt[3]{32400}\right)}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]

Alternatives

Alternative 1
Error	21.0
Cost	52352

\[{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{\sqrt[3]{180}} \cdot {\left( 3.08641975308642 \cdot 10^{-5} \right)}^{0.3333333333333333}\right)\right)}^{2} \]

Alternative 2
Error	21.0
Cost	52288

\[{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{\sqrt[3]{32400} \cdot \sqrt[3]{180}}\right)\right)}^{2} \]

Alternative 3
Error	21.0
Cost	45760

\[{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \left(\pi \cdot \sqrt[3]{1.7146776406035666 \cdot 10^{-7}}\right)\right)\right)}^{2} \]

Alternative 4
Error	21.0
Cost	39360

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

Alternative 5
Error	21.1
Cost	26240

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

Alternative 6
Error	21.5
Cost	20489

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.044 \lor \neg \left(angle \leq 1.6 \cdot 10^{-30}\right):\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \left(a \cdot angle\right)\right) \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	21.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.044 \lor \neg \left(angle \leq 1.6 \cdot 10^{-30}\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	21.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.044 \lor \neg \left(angle \leq 1.6 \cdot 10^{-30}\right):\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	26.7
Cost	19840

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

Alternative 10
Error	26.6
Cost	19840

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

Alternative 11
Error	26.6
Cost	19840

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

Alternative 12
Error	26.6
Cost	19840

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

Reproduce

herbie shell --seed 2023054 
(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)))