Maksimov and Kolovsky, Equation (3)

Average Error: 17.5 → 9.0

Time: 17.9s

Precision: binary64

Cost: 20749

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -8.5 \cdot 10^{+213}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+164} \lor \neg \left(U \leq 1.6 \cdot 10^{+188}\right):\\ \;\;\;\;\left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

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 -8.5e+213)
     U
     (if (or (<= U 1.9e+164) (not (<= U 1.6e+188)))
       (* (* t_0 (* -2.0 J)) (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 <= -8.5e+213) {
		tmp = U;
	} else if ((U <= 1.9e+164) || !(U <= 1.6e+188)) {
		tmp = (t_0 * (-2.0 * J)) * 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 <= -8.5e+213) {
		tmp = U;
	} else if ((U <= 1.9e+164) || !(U <= 1.6e+188)) {
		tmp = (t_0 * (-2.0 * J)) * 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 <= -8.5e+213:
		tmp = U
	elif (U <= 1.9e+164) or not (U <= 1.6e+188):
		tmp = (t_0 * (-2.0 * J)) * 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 <= -8.5e+213)
		tmp = U;
	elseif ((U <= 1.9e+164) || !(U <= 1.6e+188))
		tmp = Float64(Float64(t_0 * Float64(-2.0 * J)) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	else
		tmp = Float64(-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 <= -8.5e+213)
		tmp = U;
	elseif ((U <= 1.9e+164) || ~((U <= 1.6e+188)))
		tmp = (t_0 * (-2.0 * J)) * 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, -8.5e+213], U, If[Or[LessEqual[U, 1.9e+164], N[Not[LessEqual[U, 1.6e+188]], $MachinePrecision]], N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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 3 regimes

2. if U < -8.4999999999999995e213

   1. Initial program 43.0

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

   2. Simplified 27.9

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

      [Start] 43.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 43.0	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 43.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 27.9	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 27.9	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 27.9	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in U around -inf 35.1

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

   if -8.4999999999999995e213 < U < 1.90000000000000011e164 or 1.59999999999999985e188 < U

   1. Initial program 15.1

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

   2. Simplified 6.2

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

      [Start] 15.1	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 15.1	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 6.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 6.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if 1.90000000000000011e164 < U < 1.59999999999999985e188

   1. Initial program 33.3

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

   2. Simplified 18.3

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

      [Start] 33.3	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 33.4	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 33.4	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 18.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 18.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 18.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in J around 0 41.1

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

   4. Simplified 41.1

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

      [Start] 41.1	\[ -1 \cdot U \]
      mul-1-neg [=>] 41.1	\[ \color{blue}{-U} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -8.5 \cdot 10^{+213}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+164} \lor \neg \left(U \leq 1.6 \cdot 10^{+188}\right):\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternatives

Alternative 1
Error	9.0
Cost	20749

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -9 \cdot 10^{+213}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+164} \lor \neg \left(U \leq 1.6 \cdot 10^{+188}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 2
Error	21.5
Cost	7568

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;K \leq -5.1 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -5.8 \cdot 10^{+128}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -460:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{elif}\;K \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	26.6
Cost	7244

\[\begin{array}{l} \mathbf{if}\;U \leq -5.3 \cdot 10^{+175}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -3.3 \cdot 10^{+71}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 7 \cdot 10^{+163}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;U \leq 6.8 \cdot 10^{+188}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{+189}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 10^{+265}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 4
Error	47.4
Cost	1052

\[\begin{array}{l} \mathbf{if}\;U \leq -8 \cdot 10^{+178}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.6 \cdot 10^{-27}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -7.5 \cdot 10^{-73}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 4.7 \cdot 10^{+54}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 5.4 \cdot 10^{+188}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 10^{+267}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5
Error	39.4
Cost	1052

\[\begin{array}{l} \mathbf{if}\;U \leq -1.3 \cdot 10^{+179}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -5.2 \cdot 10^{+68}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.06 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq -3.3 \cdot 10^{-69}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 5.5 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 2.7 \cdot 10^{+188}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.5 \cdot 10^{+264}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 6
Error	47.2
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023045 
(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)))))