Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 8.8

Time: 19.1s

Precision: binary64

Cost: 47300

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t_0\\ \mathbf{if}\;t_1 \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \end{array} \]

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))) (t_1 (* (* -2.0 J) t_0)))
   (if (<=
        (* t_1 (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))
        (- INFINITY))
     (- U)
     (* t_1 (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 t_1 = (-2.0 * J) * t_0;
	double tmp;
	if ((t_1 * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)))) <= -((double) INFINITY)) {
		tmp = -U;
	} else {
		tmp = t_1 * 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 t_1 = (-2.0 * J) * t_0;
	double tmp;
	if ((t_1 * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)))) <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else {
		tmp = t_1 * 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))
	t_1 = (-2.0 * J) * t_0
	tmp = 0
	if (t_1 * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))) <= -math.inf:
		tmp = -U
	else:
		tmp = t_1 * 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))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	tmp = 0.0
	if (Float64(t_1 * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(-U);
	else
		tmp = Float64(t_1 * 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));
	t_1 = (-2.0 * J) * t_0;
	tmp = 0.0;
	if ((t_1 * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)))) <= -Inf)
		tmp = -U;
	else
		tmp = t_1 * 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]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], (-U), N[(t$95$1 * 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 (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in J around 0 32.4

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

   3. Simplified 32.4

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

      [Start] 32.4	\[ -1 \cdot U \]
      mul-1-neg [=>] 32.4	\[ \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 10.5

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

   2. Simplified 4.9

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

      [Start] 10.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 10.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 4.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 4.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 8.8

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

Alternatives

Alternative 1
Error	8.9
Cost	20749

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -8.4 \cdot 10^{+218} \lor \neg \left(U \leq 2.2 \cdot 10^{+251}\right) \land U \leq 8.8 \cdot 10^{+287}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	16.8
Cost	14224

\[\begin{array}{l} t_0 := \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{if}\;J \leq -8.5 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.35 \cdot 10^{-246}:\\ \;\;\;\;J \cdot \left(-2 \cdot \frac{J}{U}\right) - U\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{-200}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.6 \cdot 10^{-193}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.8
Cost	14224

\[\begin{array}{l} t_0 := \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{if}\;J \leq -2.8 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.65 \cdot 10^{-246}:\\ \;\;\;\;\frac{-2}{\frac{\frac{U}{J}}{J \cdot {\cos \left(K \cdot 0.5\right)}^{2}}} - U\\ \mathbf{elif}\;J \leq 3.3 \cdot 10^{-198}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{-192}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.5
Cost	7561

\[\begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq -5 \lor \neg \left(\frac{K}{2} \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \end{array} \]

Alternative 5
Error	25.9
Cost	7376

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -9.5 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{-246}:\\ \;\;\;\;J \cdot \left(-2 \cdot \frac{J}{U}\right) - U\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{-166}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-22}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	19.4
Cost	7305

\[\begin{array}{l} \mathbf{if}\;K \leq -0.245 \lor \neg \left(K \leq 1.8 \cdot 10^{-7}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\right)\\ \end{array} \]

Alternative 7
Error	47.4
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -1.4 \cdot 10^{+208}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.1 \cdot 10^{-144}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -1.2 \cdot 10^{-219}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 5.1 \cdot 10^{+124}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 5.2 \cdot 10^{+246}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Error	37.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;J \leq -6400000000000:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{-246}:\\ \;\;\;\;J \cdot \left(-2 \cdot \frac{J}{U}\right) - U\\ \mathbf{elif}\;J \leq 4.7 \cdot 10^{-166}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9
Error	37.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;J \leq -850000000000:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 2.9 \cdot 10^{-247}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-166}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{-12}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10
Error	47.4
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023025 
(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)))))