ab-angle->ABCF A

Average Error: 20.8 → 20.9

Time: 21.4s

Precision: binary64

Cost: 71488

Math

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

↓

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

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

Derivation

1. Initial program 20.8

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

2. Simplified 20.8

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

   [Start] 20.8	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 20.8	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 20.8	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} \]

3. Applied egg-rr 20.8

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

4. Applied egg-rr 20.9

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

5. Final simplification 20.9

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

Alternatives

Alternative 1
Error	20.8
Cost	52224

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

Alternative 2
Error	20.8
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 3
Error	20.9
Cost	45760

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

Alternative 4
Error	20.8
Cost	39040

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

Alternative 5
Error	20.9
Cost	26240

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

Alternative 6
Error	20.8
Cost	26240

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

Alternative 7
Error	21.0
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.6 \cdot 10^{+17} \lor \neg \left(angle \leq 0.0053\right):\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \frac{a}{\frac{2}{a}}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 8
Error	21.0
Cost	20424

\[\begin{array}{l} t_0 := \frac{a}{\frac{2}{a}}\\ \mathbf{if}\;angle \leq -2.6 \cdot 10^{+17}:\\ \;\;\;\;{b}^{2} + t_0 \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 0.0053:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot t_0\\ \end{array} \]

Alternative 9
Error	21.0
Cost	20424

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.6 \cdot 10^{+17}:\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 0.0053:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	21.0
Cost	20424

\[\begin{array}{l} t_0 := \frac{a \cdot a}{2}\\ \mathbf{if}\;angle \leq -2.6 \cdot 10^{+17}:\\ \;\;\;\;{b}^{2} + t_0 \cdot \left(1 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;angle \leq 0.0053:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + t_0 \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	24.2
Cost	20360

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+123}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+115}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({\pi}^{2} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 12
Error	24.4
Cost	20360

\[\begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+123}:\\ \;\;\;\;{b}^{2} + {\left(\pi \cdot \left(a \cdot angle\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({\pi}^{2} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	26.7
Cost	19840

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

Alternative 14
Error	26.7
Cost	19840

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

Alternative 15
Error	26.6
Cost	19840

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

Alternative 16
Error	26.7
Cost	19840

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

Reproduce

herbie shell --seed 2023067 
(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)))