Maksimov and Kolovsky, Equation (3)

Average Error: 17.9 → 8.4

Time: 21.6s

Precision: binary64

Cost: 20617

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -5.5 \cdot 10^{-282} \lor \neg \left(J \leq 1.2 \cdot 10^{-252}\right):\\ \;\;\;\;\left(t_0 \cdot \left(J \cdot -2\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

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 (or (<= J -5.5e-282) (not (<= J 1.2e-252)))
     (* (* t_0 (* J -2.0)) (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 ((J <= -5.5e-282) || !(J <= 1.2e-252)) {
		tmp = (t_0 * (J * -2.0)) * 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 ((J <= -5.5e-282) || !(J <= 1.2e-252)) {
		tmp = (t_0 * (J * -2.0)) * 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 (J <= -5.5e-282) or not (J <= 1.2e-252):
		tmp = (t_0 * (J * -2.0)) * 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 ((J <= -5.5e-282) || !(J <= 1.2e-252))
		tmp = Float64(Float64(t_0 * Float64(J * -2.0)) * 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));
	tmp = 0.0;
	if ((J <= -5.5e-282) || ~((J <= 1.2e-252)))
		tmp = (t_0 * (J * -2.0)) * 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[Or[LessEqual[J, -5.5e-282], N[Not[LessEqual[J, 1.2e-252]], $MachinePrecision]], N[(N[(t$95$0 * N[(J * -2.0), $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 2 regimes

2. if J < -5.5000000000000001e-282 or 1.2000000000000001e-252 < J

   1. Initial program 16.1

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

   2. Simplified 6.5

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

      [Start] 16.1	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 16.1	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 6.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 6.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if -5.5000000000000001e-282 < J < 1.2000000000000001e-252

   1. Initial program 44.5

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

   2. Simplified 27.7

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

      [Start] 44.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 44.5	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 44.5	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 27.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 27.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 27.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in U around -inf 35.9

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -5.5 \cdot 10^{-282} \lor \neg \left(J \leq 1.2 \cdot 10^{-252}\right):\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternatives

Alternative 1
Error	8.5
Cost	20617

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -3.6 \cdot 10^{-282} \lor \neg \left(J \leq 1.05 \cdot 10^{-252}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2
Error	8.5
Cost	20617

\[\begin{array}{l} \mathbf{if}\;J \leq -5 \cdot 10^{-282} \lor \neg \left(J \leq 3.4 \cdot 10^{-251}\right):\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J}}{2 \cdot \cos \left(K \cdot 0.5\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 3
Error	17.9
Cost	14884

\[\begin{array}{l} t_0 := \frac{J}{U} \cdot \left(J \cdot -2\right) - U\\ t_1 := -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\right)\\ \mathbf{if}\;J \leq -1.1 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.3 \cdot 10^{-122}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -3.2 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.7 \cdot 10^{-201}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-242}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -6 \cdot 10^{-278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{-262}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.02 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{-185}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	23.2
Cost	8228

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot -2\right)\\ t_1 := -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right)\right)\\ t_2 := \frac{J}{U} \cdot \left(J \cdot -2\right) - U\\ \mathbf{if}\;J \leq -6.6 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -5.2 \cdot 10^{-241}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.1 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2.8 \cdot 10^{-281}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{-261}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq 5 \cdot 10^{-185}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.32 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	25.9
Cost	7640

\[\begin{array}{l} t_0 := \frac{J}{U} \cdot \left(J \cdot -2\right) - U\\ t_1 := \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;J \leq -3.9 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -3.2 \cdot 10^{-126}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -6.2 \cdot 10^{-281}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 3.8 \cdot 10^{-262}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.1 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.35 \cdot 10^{-104}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	38.2
Cost	1236

\[\begin{array}{l} t_0 := \frac{J}{U} \cdot \left(J \cdot -2\right) - U\\ \mathbf{if}\;J \leq -2.7 \cdot 10^{-94}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -1.46 \cdot 10^{-126}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.1 \cdot 10^{-282}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-262}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.16 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-63}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 7
Error	38.2
Cost	984

\[\begin{array}{l} \mathbf{if}\;J \leq -3.8 \cdot 10^{-94}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.8 \cdot 10^{-281}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-262}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-218}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.22 \cdot 10^{-63}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 8
Error	46.4
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -2.5 \cdot 10^{+171}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.15 \cdot 10^{+30}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -1.95 \cdot 10^{-26}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -3.5 \cdot 10^{-189}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.25 \cdot 10^{-181}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 7.6 \cdot 10^{+18}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Error	46.8
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022354 
(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)))))