Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 13.6

Time: 28.4s

Precision: binary64

Cost: 80776

Math

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

↓

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

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 INFINITY) 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 <= ((double) INFINITY)) {
		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 <= Double.POSITIVE_INFINITY) {
		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 <= math.inf:
		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 = U;
	elseif (t_1 <= Inf)
		tmp = t_1;
	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));
	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 <= Inf)
		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, Infinity], 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. Taylor expanded in U around -inf 30.8

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

   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)))) < +inf.0

   1. Initial program 10.7

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

   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 18.4

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

   2. Taylor expanded in J around 0 46.8

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

   3. Simplified 46.8

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

      [Start] 46.8	\[ -1 \cdot U \]
      rational.json-simplify-2 [=>] 46.8	\[ \color{blue}{U \cdot -1} \]
      rational.json-simplify-9 [=>] 46.8	\[ \color{blue}{-U} \]

3. Recombined 3 regimes into one program.

4. Final simplification 13.6

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

Alternatives

Alternative 1
Error	17.3
Cost	27144

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;U \leq -6.2 \cdot 10^{+141}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 10^{+242}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{1 + {\left(U \cdot \frac{0.5}{J \cdot t_0}\right)}^{2}} \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 2
Error	23.4
Cost	20488

\[\begin{array}{l} \mathbf{if}\;U \leq -6.8 \cdot 10^{-13}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{+237}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J + J}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 3
Error	27.1
Cost	14624

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U}{J + J}\right)}^{2}}\\ t_1 := \cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;J \leq -2.65 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2.1 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.3 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -4.4 \cdot 10^{-136}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -3.9 \cdot 10^{-266}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-261}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-205}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 13.2:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \cos \left(-0.5 \cdot K\right) \cdot \left(J + J\right)\right)\\ \end{array} \]

Alternative 4
Error	27.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;U \leq -1.75 \cdot 10^{-11}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 8000000000000:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;U \leq 10^{+95}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.1 \cdot 10^{+177}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{+236}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5
Error	39.0
Cost	1248

\[\begin{array}{l} \mathbf{if}\;J \leq -2.05 \cdot 10^{+81}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -7.6 \cdot 10^{-21}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2.9 \cdot 10^{-132}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.7 \cdot 10^{-266}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4 \cdot 10^{-261}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5 \cdot 10^{-209}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{+64}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 6
Error	47.1
Cost	524

\[\begin{array}{l} \mathbf{if}\;K \leq -1.12 \cdot 10^{-259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 4.6 \cdot 10^{-160}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 8 \cdot 10^{-43}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 7
Error	46.6
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023064 
(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)))))