Maksimov and Kolovsky, Equation (3)

Average Error: 18.6 → 8.5

Time: 19.8s

Precision: binary64

Cost: 20484

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -1.95 \cdot 10^{+264}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \end{array} \]

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 -1.95e+264)
     (- U)
     (* (* (* -2.0 J) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0))))))))

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 <= -1.95e+264) {
		tmp = -U;
	} else {
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	}
	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 <= -1.95e+264) {
		tmp = -U;
	} else {
		tmp = ((-2.0 * J) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	}
	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 <= -1.95e+264:
		tmp = -U
	else:
		tmp = ((-2.0 * J) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	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 <= -1.95e+264)
		tmp = Float64(-U);
	else
		tmp = Float64(Float64(Float64(-2.0 * J) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	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 <= -1.95e+264)
		tmp = -U;
	else
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	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, -1.95e+264], (-U), 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]]]

Derivation

1. Split input into 2 regimes

2. if U < -1.94999999999999997e264

   1. Initial program 47.9

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

   2. Simplified 47.9

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

      [Start] 47.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 47.9	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      associate-*l* [=>] 47.9	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right)}^{2}}\right) \]

   3. Taylor expanded in J around 0 35.3

      \[\leadsto \color{blue}{-1 \cdot U} \]

   4. Simplified 35.3

      \[\leadsto \color{blue}{-U} \]
      Proof

      [Start] 35.3	\[ -1 \cdot U \]
      mul-1-neg [=>] 35.3	\[ \color{blue}{-U} \]

   if -1.94999999999999997e264 < U

   1. Initial program 17.4

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

   2. Simplified 7.5

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

      [Start] 17.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 17.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 7.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 7.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.95 \cdot 10^{+264}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.5
Cost	20484

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -3 \cdot 10^{+272}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right)\\ \end{array} \]

Alternative 2
Error	17.2
Cost	13960

\[\begin{array}{l} \mathbf{if}\;U \leq -5.8 \cdot 10^{+240}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3 \cdot 10^{+248}:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 3
Error	17.3
Cost	13960

\[\begin{array}{l} \mathbf{if}\;U \leq -3.5 \cdot 10^{+242}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.75 \cdot 10^{+248}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 4
Error	24.2
Cost	7832

\[\begin{array}{l} t_0 := J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ t_1 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -7.5 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.92 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 4 \cdot 10^{-247}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 8 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 4.4 \cdot 10^{-38}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	27.1
Cost	7508

\[\begin{array}{l} \mathbf{if}\;U \leq -3.6 \cdot 10^{+262}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.2 \cdot 10^{+104}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -3.3 \cdot 10^{+78}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -30000000:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.05 \cdot 10^{+82}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U \leq 2.15 \cdot 10^{+278}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 6
Error	47.1
Cost	1184

\[\begin{array}{l} \mathbf{if}\;U \leq -4 \cdot 10^{+262}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.6 \cdot 10^{+78}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -800000:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.3 \cdot 10^{-83}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.1 \cdot 10^{+88}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{+104}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3 \cdot 10^{+276}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7
Error	38.5
Cost	920

\[\begin{array}{l} \mathbf{if}\;U \leq -4 \cdot 10^{+262}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.12 \cdot 10^{+104}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -7.8 \cdot 10^{+77}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -27000000:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 10^{+278}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Error	46.7
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023011 
(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)))))