Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 9.4

Time: 1.1min

Precision: binary64

Cost: 80776

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

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)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U) (if (<= t_1 4e+306) t_1 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 t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 4e+306) {
		tmp = t_1;
	} 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 t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 4e+306) {
		tmp = t_1;
	} 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))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 4e+306:
		tmp = t_1
	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))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 4e+306)
		tmp = t_1;
	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));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 4e+306)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 4e+306], t$95$1, U]]]]

Derivation

1. Split input into 3 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. Simplified 63.9

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

      [Start] 64.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      rational_best-simplify-1 [=>] 64.0	\[ \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \]
      rational_best-simplify-1 [=>] 64.0	\[ \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)} \]
      rational_best-simplify-50 [=>] 63.9	\[ \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      rational_best-simplify-1 [=>] 63.9	\[ \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \color{blue}{\left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J\right)} \]
      rational_best-simplify-50 [=>] 63.9	\[ \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      rational_best-simplify-54 [=>] 63.9	\[ J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{\frac{U}{2 \cdot J}}{\cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right) \]
      rational_best-simplify-49 [=>] 63.9	\[ J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{\frac{U}{\cos \left(\frac{K}{2}\right)}}{2 \cdot J}\right)}}^{2}}\right) \]
      rational_best-simplify-1 [=>] 63.9	\[ J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{\frac{U}{\cos \left(\frac{K}{2}\right)}}{\color{blue}{J \cdot 2}}\right)}^{2}}\right) \]

   3. Taylor expanded in J around 0 32.9

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

   4. Simplified 32.9

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

      [Start] 32.9	\[ -1 \cdot U \]
      rational_best-simplify-1 [=>] 32.9	\[ \color{blue}{U \cdot -1} \]
      rational_best-simplify-11 [<=] 32.9	\[ \color{blue}{-U} \]

   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)))) < 4.00000000000000007e306

   1. Initial program 0.1

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

   if 4.00000000000000007e306 < (*.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 63.0

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

   2. Simplified 63.0

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

      [Start] 63.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      rational_best-simplify-1 [=>] 63.0	\[ \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \]
      rational_best-simplify-1 [=>] 63.0	\[ \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)} \]
      rational_best-simplify-50 [=>] 63.0	\[ \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      rational_best-simplify-1 [=>] 63.0	\[ \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \color{blue}{\left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J\right)} \]
      rational_best-simplify-50 [=>] 63.0	\[ \color{blue}{J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      rational_best-simplify-54 [=>] 63.0	\[ J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{\frac{U}{2 \cdot J}}{\cos \left(\frac{K}{2}\right)}\right)}}^{2}}\right) \]
      rational_best-simplify-49 [=>] 63.0	\[ J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{\frac{U}{\cos \left(\frac{K}{2}\right)}}{2 \cdot J}\right)}}^{2}}\right) \]
      rational_best-simplify-1 [=>] 63.0	\[ J \cdot \left(\left(-2 \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{\frac{U}{\cos \left(\frac{K}{2}\right)}}{\color{blue}{J \cdot 2}}\right)}^{2}}\right) \]

   3. Taylor expanded in U around -inf 32.1

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternatives

Alternative 1
Error	17.1
Cost	27276

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\left(-2 \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{\frac{U}{t_0}}{J \cdot 2}\right)}^{2}}\right)\\ \mathbf{if}\;J \leq -1.35 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 3 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-125}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	23.4
Cost	20620

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \left(\sqrt{1 + {\left(U \cdot \frac{0.5}{J}\right)}^{2}} \cdot \cos \left(0.5 \cdot K\right)\right)\\ \mathbf{if}\;J \leq -1.15 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.35 \cdot 10^{+21}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	23.4
Cost	20620

\[\begin{array}{l} \mathbf{if}\;J \leq -1.5 \cdot 10^{-213}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J + J}\right)}^{2}}\\ \mathbf{elif}\;J \leq 1.08 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.35 \cdot 10^{+21}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\sqrt{1 + {\left(U \cdot \frac{0.5}{J}\right)}^{2}} \cdot \cos \left(0.5 \cdot K\right)\right)\\ \end{array} \]

Alternative 4
Error	27.1
Cost	13832

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;J \leq -4.6 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2.5 \cdot 10^{-167}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{1 + {\left(\frac{U}{J + J}\right)}^{2}}\\ \mathbf{elif}\;J \leq -2.4 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.4 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.15 \cdot 10^{+23}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	27.0
Cost	7244

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;J \leq -3.6 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 3.8 \cdot 10^{-305}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	38.1
Cost	588

\[\begin{array}{l} \mathbf{if}\;J \leq -2 \cdot 10^{-59}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq 3 \cdot 10^{-306}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3 \cdot 10^{+77}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 7
Error	47.0
Cost	524

\[\begin{array}{l} \mathbf{if}\;K \leq -1.3 \cdot 10^{-202}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -6.6 \cdot 10^{-268}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.5 \cdot 10^{-270}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	46.5
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023099 
(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)))))