Maksimov and Kolovsky, Equation (3)

Average Error: 28.76% → 13.23%

Time: 18.5s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -6.6 \cdot 10^{+259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{+258}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+297}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

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 (<= U -6.6e+259)
     (- U)
     (if (<= U 2.8e+258)
       (* (* (* -2.0 J) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))
       (if (<= U 5e+297) U (- 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 (U <= -6.6e+259) {
		tmp = -U;
	} else if (U <= 2.8e+258) {
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	} else if (U <= 5e+297) {
		tmp = U;
	} 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 (U <= -6.6e+259) {
		tmp = -U;
	} else if (U <= 2.8e+258) {
		tmp = ((-2.0 * J) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	} else if (U <= 5e+297) {
		tmp = U;
	} 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 U <= -6.6e+259:
		tmp = -U
	elif U <= 2.8e+258:
		tmp = ((-2.0 * J) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	elif U <= 5e+297:
		tmp = U
	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 (U <= -6.6e+259)
		tmp = Float64(-U);
	elseif (U <= 2.8e+258)
		tmp = Float64(Float64(Float64(-2.0 * J) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	elseif (U <= 5e+297)
		tmp = U;
	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));
	tmp = 0.0;
	if (U <= -6.6e+259)
		tmp = -U;
	elseif (U <= 2.8e+258)
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	elseif (U <= 5e+297)
		tmp = U;
	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[LessEqual[U, -6.6e+259], (-U), If[LessEqual[U, 2.8e+258], N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 5e+297], U, (-U)]]]]

Derivation

1. Split input into 3 regimes

2. if U < -6.6000000000000001e259 or 4.9999999999999998e297 < U

   1. Initial program 69.83

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

   2. Taylor expanded in J around 0 50.98

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

   3. Simplified 50.98

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

      [Start] 50.98	\[ -1 \cdot U \]
      mul-1-neg [=>] 50.98	\[ \color{blue}{-U} \]

   if -6.6000000000000001e259 < U < 2.79999999999999982e258

   1. Initial program 25

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

   2. Simplified 9.63

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

      [Start] 25	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 25	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 9.63	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 9.63	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if 2.79999999999999982e258 < U < 4.9999999999999998e297

   1. Initial program 68.17

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

   2. Taylor expanded in U around -inf 53.44

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

4. Final simplification 13.23

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -6.6 \cdot 10^{+259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{+258}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+297}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternatives

Alternative 1
Error	27.01%
Cost	13960

\[\begin{array}{l} \mathbf{if}\;U \leq -5.5 \cdot 10^{+259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 7 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+196}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 10^{+258}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2
Error	37.58%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{+256}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.4 \cdot 10^{-37}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\\ \mathbf{elif}\;U \leq 6.2 \cdot 10^{+57}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]

Alternative 3
Error	37.56%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;U \leq -3.2 \cdot 10^{+256}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.75 \cdot 10^{-36}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\\ \mathbf{elif}\;U \leq 4.5 \cdot 10^{+57}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]

Alternative 4
Error	41.43%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;U \leq -6.6 \cdot 10^{+259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -5.8 \cdot 10^{+76}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]

Alternative 5
Error	59.86%
Cost	1236

\[\begin{array}{l} \mathbf{if}\;U \leq -4.8 \cdot 10^{+259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.8 \cdot 10^{+53}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+149}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{+196}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]

Alternative 6
Error	72.62%
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -3.9 \cdot 10^{+90}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -5.6 \cdot 10^{-53}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.2 \cdot 10^{-129}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -1.6 \cdot 10^{-276}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 4.6 \cdot 10^{+45}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.55 \cdot 10^{+161}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7
Error	59.89%
Cost	788

\[\begin{array}{l} \mathbf{if}\;U \leq -4.2 \cdot 10^{+259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -3.4 \cdot 10^{+76}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.5 \cdot 10^{+51}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{+149}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{+197}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Error	73.88%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023090 
(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)))))