ab-angle->ABCF A

Average Error: 20.5 → 20.6

Time: 21.7s

Precision: binary64

Cost: 104000

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{180}{\pi}}\\ t_1 := \sqrt[3]{t_0}\\ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{angle}{t_1 \cdot \left(t_0 \cdot {t_1}^{2}\right)}}{t_0}\right)\right)}^{2} \end{array} \]

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 (/ 180.0 PI))) (t_1 (cbrt t_0)))
   (+
    (pow (* a (sin (/ angle (/ 180.0 PI)))) 2.0)
    (pow (* b (cos (/ (/ angle (* t_1 (* t_0 (pow t_1 2.0)))) t_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) {
	double t_0 = cbrt((180.0 / ((double) M_PI)));
	double t_1 = cbrt(t_0);
	return pow((a * sin((angle / (180.0 / ((double) M_PI))))), 2.0) + pow((b * cos(((angle / (t_1 * (t_0 * pow(t_1, 2.0)))) / t_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) {
	double t_0 = Math.cbrt((180.0 / Math.PI));
	double t_1 = Math.cbrt(t_0);
	return Math.pow((a * Math.sin((angle / (180.0 / Math.PI)))), 2.0) + Math.pow((b * Math.cos(((angle / (t_1 * (t_0 * Math.pow(t_1, 2.0)))) / t_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)
	t_0 = cbrt(Float64(180.0 / pi))
	t_1 = cbrt(t_0)
	return Float64((Float64(a * sin(Float64(angle / Float64(180.0 / pi)))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / Float64(t_1 * Float64(t_0 * (t_1 ^ 2.0)))) / t_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_] := Block[{t$95$0 = N[Power[N[(180.0 / Pi), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 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$1 * N[(t$95$0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 20.5

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

2. Simplified 20.5

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

   [Start] 20.5	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 20.5	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   associate-/r/ [<=] 20.5	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle}{\frac{180}{\pi}}\right)}\right)}^{2} \]

3. Applied egg-rr 20.5

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

4. Simplified 20.6

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

   [Start] 20.5	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{1}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}} \cdot \frac{angle}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   associate-*r/ [=>] 20.6	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{\frac{1}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}} \cdot angle}{\sqrt[3]{\frac{180}{\pi}}}\right)}\right)}^{2} \]
   associate-*l/ [=>] 20.6	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\frac{1 \cdot angle}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]
   *-lft-identity [=>] 20.6	\[ {\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\frac{\color{blue}{angle}}{{\left(\sqrt[3]{\frac{180}{\pi}}\right)}^{2}}}{\sqrt[3]{\frac{180}{\pi}}}\right)\right)}^{2} \]

5. Applied egg-rr 20.6

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

6. Final simplification 20.6

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

Alternatives

Alternative 1
Error	20.6
Cost	52224

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

Alternative 2
Error	20.5
Cost	39360

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ {\left(b \cdot \cos t_0\right)}^{2} + {\left(a \cdot \sin t_0\right)}^{2} \end{array} \]

Alternative 3
Error	20.5
Cost	39360

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

Alternative 4
Error	20.5
Cost	39360

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

Alternative 5
Error	20.5
Cost	39360

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

Alternative 6
Error	20.6
Cost	26240

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

Alternative 7
Error	20.5
Cost	26240

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

Alternative 8
Error	20.6
Cost	26240

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

Alternative 9
Error	20.6
Cost	26240

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

Alternative 10
Error	20.7
Cost	20745

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \lor \neg \left(\frac{angle}{180} \leq 0.5\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(a \cdot angle\right)\\ \end{array} \]

Alternative 11
Error	20.7
Cost	20681

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \lor \neg \left(\frac{angle}{180} \leq 0.5\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 12
Error	20.7
Cost	20681

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \lor \neg \left(\frac{angle}{180} \leq 0.5\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 13
Error	22.5
Cost	20420

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -0.0005:\\ \;\;\;\;\frac{a}{\frac{2}{a}} \cdot 0 + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{+58}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]

Alternative 14
Error	22.4
Cost	20105

\[\begin{array}{l} \mathbf{if}\;angle \leq -8.2 \cdot 10^{+42} \lor \neg \left(angle \leq 2.9 \cdot 10^{+60}\right):\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 15
Error	22.3
Cost	20105

\[\begin{array}{l} \mathbf{if}\;angle \leq -6 \cdot 10^{+42} \lor \neg \left(angle \leq 2.9 \cdot 10^{+60}\right):\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 16
Error	22.3
Cost	19977

\[\begin{array}{l} \mathbf{if}\;angle \leq -6 \cdot 10^{+42} \lor \neg \left(angle \leq 2.9 \cdot 10^{+60}\right):\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b\right)\right)}^{2}\\ \end{array} \]

Alternative 17
Error	30.6
Cost	14025

\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+149} \lor \neg \left(a \leq -1.46 \cdot 10^{-40}\right):\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left({\pi}^{2} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot angle\right)\right)\\ \end{array} \]

Alternative 18
Error	32.1
Cost	192

\[b \cdot b \]

Reproduce

herbie shell --seed 2023041 
(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)))