ab-angle->ABCF A

Average Accuracy: 68.2% → 68.3%

Time: 23.7s

Precision: binary64

Cost: 71680

Math

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

↓

\[{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\left|\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right| + 1}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]

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
    (pow
     (cbrt
      (/
       (sqrt (+ (fabs (cos (* PI (* angle 0.011111111111111112)))) 1.0))
       (sqrt 2.0)))
     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 / (180.0 / ((double) M_PI))))), 2.0) + pow((b * pow(cbrt((sqrt((fabs(cos((((double) M_PI) * (angle * 0.011111111111111112)))) + 1.0)) / sqrt(2.0))), 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 / (180.0 / Math.PI)))), 2.0) + Math.pow((b * Math.pow(Math.cbrt((Math.sqrt((Math.abs(Math.cos((Math.PI * (angle * 0.011111111111111112)))) + 1.0)) / Math.sqrt(2.0))), 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(180.0 / pi)))) ^ 2.0) + (Float64(b * (cbrt(Float64(sqrt(Float64(abs(cos(Float64(pi * Float64(angle * 0.011111111111111112)))) + 1.0)) / sqrt(2.0))) ^ 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[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Power[N[Power[N[(N[Sqrt[N[(N[Abs[N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

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

2. Simplified 68.3%

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

   [Start] 68.2	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 68.3	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} \]

3. Applied egg-rr 68.3%

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

   [Start] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} \]
   add-cube-cbrt [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)} \cdot \sqrt[3]{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}\right)}^{2} \]
   pow3 [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \color{blue}{{\left(\sqrt[3]{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{3}}\right)}^{2} \]
   associate-/l* [<=] 68.2	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}}\right)}^{3}\right)}^{2} \]
   associate-*r/ [<=] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}}\right)}^{3}\right)}^{2} \]
   div-inv [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\cos \left(angle \cdot \color{blue}{\left(\pi \cdot \frac{1}{180}\right)}\right)}\right)}^{3}\right)}^{2} \]
   metadata-eval [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot \color{blue}{0.005555555555555556}\right)\right)}\right)}^{3}\right)}^{2} \]

4. Applied egg-rr 68.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) + \cos 0}}{\sqrt{2}}}}\right)}^{3}\right)}^{2} \]
   Proof

   [Start] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)}^{2} \]
   rem-cube-cbrt [<=] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\color{blue}{{\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}}}\right)}^{3}\right)}^{2} \]
   add-sqr-sqrt [=>] 61.9	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}} \cdot \sqrt{{\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}}}}\right)}^{3}\right)}^{2} \]
   sqrt-unprod [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3} \cdot {\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}}}}\right)}^{3}\right)}^{2} \]
   rem-cube-cbrt [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\sqrt{\color{blue}{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)} \cdot {\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}}}\right)}^{3}\right)}^{2} \]
   rem-cube-cbrt [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\sqrt{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \color{blue}{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}}\right)}^{3}\right)}^{2} \]
   cos-mult [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\sqrt{\color{blue}{\frac{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) - angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}{2}}}}\right)}^{3}\right)}^{2} \]
   sqrt-div [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) - angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}{\sqrt{2}}}}\right)}^{3}\right)}^{2} \]

5. Applied egg-rr 68.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{2}}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   Proof

   [Start] 68.2	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   add-sqr-sqrt [=>] 62.0	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{\sqrt{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \cdot \sqrt{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   sqrt-unprod [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{\sqrt{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   pow2 [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\sqrt{\color{blue}{{\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}^{2}}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   *-commutative [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\sqrt{{\cos \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.011111111111111112\right)}^{2}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   associate-*l* [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\sqrt{{\cos \color{blue}{\left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}}^{2}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]

6. Simplified 68.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right|} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   Proof

   [Start] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}^{2}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   unpow2 [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\sqrt{\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]
   rem-sqrt-square [=>] 68.3	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{\left|\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right|} + \cos 0}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]

7. Final simplification 68.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot {\left(\sqrt[3]{\frac{\sqrt{\left|\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right| + 1}}{\sqrt{2}}}\right)}^{3}\right)}^{2} \]

Alternatives

Alternative 1
Accuracy	68.2%
Cost	65344

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

Alternative 2
Accuracy	68.2%
Cost	39488

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

Alternative 3
Accuracy	68.1%
Cost	26240

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

Alternative 4
Accuracy	68.2%
Cost	26240

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

Alternative 5
Accuracy	68.1%
Cost	26240

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

Alternative 6
Accuracy	68.1%
Cost	26240

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

Alternative 7
Accuracy	67.9%
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.005 \lor \neg \left(angle \leq 0.0046\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 8
Accuracy	62.8%
Cost	20360

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-20}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+137}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({\pi}^{2} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]

Alternative 9
Accuracy	59.0%
Cost	19840

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

Alternative 10
Accuracy	59.0%
Cost	19840

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

Reproduce

herbie shell --seed 2023130 
(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)))