Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 8.8

Time: 17.6s

Precision: binary64

Cost: 20484

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 1.56 \cdot 10^{+230}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* -2.0 J) (cos (/ K 2.0)))
  (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))

↓

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U 1.56e+230)
     (* (* (* -2.0 J) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))
     U)))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + pow((U / ((2.0 * J) * cos((K / 2.0)))), 2.0)));
}

↓

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 1.56e+230) {
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	return ((-2.0 * J) * Math.cos((K / 2.0))) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * Math.cos((K / 2.0)))), 2.0)));
}

↓

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U <= 1.56e+230) {
		tmp = ((-2.0 * J) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * math.cos((K / 2.0)))), 2.0)))

↓

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= 1.56e+230:
		tmp = ((-2.0 * J) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	else:
		tmp = U
	return tmp

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * cos(Float64(K / 2.0)))) ^ 2.0))))
end

↓

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= 1.56e+230)
		tmp = Float64(Float64(Float64(-2.0 * J) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + ((U / ((2.0 * J) * cos((K / 2.0)))) ^ 2.0)));
end

↓

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U <= 1.56e+230)
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, 1.56e+230], N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if U < 1.5599999999999999e230

   1. Initial program 16.7

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Simplified 7.0

      \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]
      Proof

      [Start] 16.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 16.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 7.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 7.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if 1.5599999999999999e230 < U

   1. Initial program 44.9

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Simplified 27.8

      \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \]
      Proof

      [Start] 44.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 44.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 27.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 27.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Taylor expanded in U around -inf 36.2

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.56 \cdot 10^{+230}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternatives

Alternative 1
Error	8.8
Cost	20484

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 7.9 \cdot 10^{+229}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2
Error	16.3
Cost	14620

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot t_0\\ \mathbf{if}\;J \leq -1.35 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -9.8 \cdot 10^{-232}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -4.1 \cdot 10^{-246}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -8.8 \cdot 10^{-256}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot t_0\\ \mathbf{elif}\;J \leq 10^{-236}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{-210}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 9 \cdot 10^{-209}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	25.5
Cost	7376

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -8.5 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{-235}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.35 \cdot 10^{-212}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-102}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	20.6
Cost	7304

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;K \leq -0.00047:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq 7200:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{elif}\;K \leq 1.15 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5
Error	37.0
Cost	1100

\[\begin{array}{l} \mathbf{if}\;J \leq -0.0058:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{-237}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.3 \cdot 10^{-211}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - J \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 5.5 \cdot 10^{-43}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 6
Error	46.9
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -2.55 \cdot 10^{-176}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 2.3 \cdot 10^{-302}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 0.33:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 8.8 \cdot 10^{+145}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.55 \cdot 10^{+211}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7
Error	37.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;J \leq -0.0058:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{-237}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.3 \cdot 10^{-211}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.4 \cdot 10^{-43}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8
Error	47.4
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023012 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))