Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 8.7

Time: 20.3s

Precision: binary64

Cost: 47300

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t_0\\ \mathbf{if}\;t_1 \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t_1 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

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))) (t_1 (* (* -2.0 J) t_0)))
   (if (<= (* t_1 (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0)))) 5e+304)
     (* t_1 (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 t_1 = (-2.0 * J) * t_0;
	double tmp;
	if ((t_1 * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)))) <= 5e+304) {
		tmp = t_1 * 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 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)))) <= 5e+304) {
		tmp = t_1 * 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))
	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)))) <= 5e+304:
		tmp = t_1 * 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))
	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)))) <= 5e+304)
		tmp = Float64(t_1 * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	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;
	tmp = 0.0;
	if ((t_1 * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)))) <= 5e+304)
		tmp = t_1 * 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]}, 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], 5e+304], 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], U]]]

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)))) < 4.9999999999999997e304

   1. Initial program 10.7

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

   2. Simplified 4.8

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

      [Start] 10.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 10.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 4.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 4.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if 4.9999999999999997e304 < (*.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 62.3

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

   2. Simplified 27.8

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

      [Start] 62.3	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 62.3	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 27.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 27.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Taylor expanded in U around -inf 32.1

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternatives

Alternative 1
Error	8.5
Cost	20617

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.15 \cdot 10^{-214} \lor \neg \left(J \leq -6.8 \cdot 10^{-264}\right):\\ \;\;\;\;-2 \cdot \left(J \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{t_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2
Error	8.5
Cost	20617

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -1.15 \cdot 10^{-214} \lor \neg \left(J \leq -1.86 \cdot 10^{-268}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 3
Error	26.8
Cost	14688

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ t_1 := -2 \cdot \left(J \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U}{J} \cdot \frac{U}{J}, 1\right)}\right)\\ \mathbf{if}\;K \leq -2.3 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -3.8 \cdot 10^{+34}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -4.9 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -1.8 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq -1.65 \cdot 10^{-273}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.8 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;K \leq 4.4 \cdot 10^{-141}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 0.018:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	16.8
Cost	14224

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{if}\;J \leq -1.15 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.2 \cdot 10^{-263}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-174}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 7.4 \cdot 10^{-170}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	16.8
Cost	14224

\[\begin{array}{l} t_0 := \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{if}\;J \leq -1.15 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 6.5 \cdot 10^{-264}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.8 \cdot 10^{-175}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.8 \cdot 10^{-170}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	26.5
Cost	7376

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.1 \cdot 10^{-214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 5.2 \cdot 10^{-265}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.55 \cdot 10^{-174}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5 \cdot 10^{-64}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	46.8
Cost	1052

\[\begin{array}{l} \mathbf{if}\;K \leq -1.5 \cdot 10^{+32}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.3 \cdot 10^{-21}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -8.5 \cdot 10^{-187}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -5.7 \cdot 10^{-216}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 5.7 \cdot 10^{+249}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.5 \cdot 10^{+275}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Error	38.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;J \leq -4.2 \cdot 10^{-94}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-264}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.3 \cdot 10^{-175}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.2 \cdot 10^{+26}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9
Error	46.7
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022356 
(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)))))