ab-angle->ABCF A

Average Error: 20.2 → 20.2

Time: 18.6s

Precision: binary64

Cost: 65088

Math

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

↓

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

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) (/ 180.0 angle)) (pow (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)) / (180.0 / angle)) * pow(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) / (180.0 / angle)) * Math.pow(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(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(cbrt(pi) / Float64(180.0 / angle)) * (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[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(N[Power[Pi, 1/3], $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.2

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

2. Applied egg-rr 20.2

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

3. Simplified 20.2

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{\sqrt[3]{\pi}}{\frac{180}{angle}} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)}\right)}^{2} \]
   Proof
   (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 (cbrt.f64 (PI.f64)) (/.f64 180 angle)) (pow.f64 (cbrt.f64 (PI.f64)) 2)))) 2)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (cbrt.f64 (PI.f64)) (/.f64 (/.f64 180 angle) (pow.f64 (cbrt.f64 (PI.f64)) 2)))))) 2)): 0 points increase in error, 2 points decrease in error

4. Final simplification 20.2

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

Alternatives

Alternative 1
Error	20.2
Cost	26240

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

Alternative 2
Error	20.2
Cost	26240

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

Alternative 3
Error	20.3
Cost	26240

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

Alternative 4
Error	20.3
Cost	26240

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

Alternative 5
Error	20.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0045 \lor \neg \left(angle \leq 9.5 \cdot 10^{-22}\right):\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 6
Error	20.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0045 \lor \neg \left(angle \leq 9.5 \cdot 10^{-22}\right):\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 7
Error	20.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0045 \lor \neg \left(angle \leq 9.5 \cdot 10^{-22}\right):\\ \;\;\;\;{b}^{2} + \frac{1 - \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{\frac{2}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 8
Error	20.6
Cost	20424

\[\begin{array}{l} t_0 := 1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;angle \leq -0.0045:\\ \;\;\;\;{b}^{2} + \frac{a}{\frac{2}{a}} \cdot t_0\\ \mathbf{elif}\;angle \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot t_0\\ \end{array} \]

Alternative 9
Error	23.5
Cost	20360

\[\begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+137}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-162}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({\pi}^{2} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 10
Error	25.8
Cost	19840

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

Alternative 11
Error	25.7
Cost	19840

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

Alternative 12
Error	25.7
Cost	19840

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

Alternative 13
Error	25.7
Cost	19840

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

Reproduce

herbie shell --seed 2022343 
(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)))