Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 8.3

Time: 22.0s

Precision: binary64

Cost: 20617

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -2.05 \cdot 10^{-302} \lor \neg \left(J \leq 8 \cdot 10^{-287}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

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 -2.05e-302) (not (<= J 8e-287)))
     (* (* (* J -2.0) t_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 <= -2.05e-302) || !(J <= 8e-287)) {
		tmp = ((J * -2.0) * t_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 <= -2.05e-302) || !(J <= 8e-287)) {
		tmp = ((J * -2.0) * t_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 <= -2.05e-302) or not (J <= 8e-287):
		tmp = ((J * -2.0) * t_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 <= -2.05e-302) || !(J <= 8e-287))
		tmp = Float64(Float64(Float64(J * -2.0) * t_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 <= -2.05e-302) || ~((J <= 8e-287)))
		tmp = ((J * -2.0) * t_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, -2.05e-302], N[Not[LessEqual[J, 8e-287]], $MachinePrecision]], N[(N[(N[(J * -2.0), $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], U]]

Derivation

1. Split input into 2 regimes

2. if J < -2.0499999999999999e-302 or 8.00000000000000017e-287 < J

   1. Initial program 17.7

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

   2. Simplified 7.7

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

      [Start] 17.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 17.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 7.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 7.7	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if -2.0499999999999999e-302 < J < 8.00000000000000017e-287

   1. Initial program 49.1

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

   2. Simplified 35.1

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

      [Start] 49.1	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 49.2	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 49.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 35.1	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 35.1	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 35.1	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in U around -inf 31.4

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.05 \cdot 10^{-302} \lor \neg \left(J \leq 8 \cdot 10^{-287}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternatives

Alternative 1
Error	8.4
Cost	20617

\[\begin{array}{l} \mathbf{if}\;J \leq -2.3 \cdot 10^{-302} \lor \neg \left(J \leq 2.8 \cdot 10^{-286}\right):\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2
Error	8.4
Cost	20616

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -2.1 \cdot 10^{-302}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{elif}\;J \leq 8.5 \cdot 10^{-287}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	8.3
Cost	20616

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -2.3 \cdot 10^{-302}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-286}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right)\\ \end{array} \]

Alternative 4
Error	18.9
Cost	14620

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -1.02 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2.35 \cdot 10^{-302}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.15 \cdot 10^{-274}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7 \cdot 10^{-266}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.65 \cdot 10^{-87}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3 \cdot 10^{-39}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	18.8
Cost	14620

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -6.3 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2.5 \cdot 10^{-302}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{-272}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.12 \cdot 10^{-265}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.65 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.9
Cost	7833

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;K \leq -4.4 \cdot 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -1.52 \cdot 10^{+187}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;K \leq -1.54 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -1.45 \cdot 10^{+83}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -0.225 \lor \neg \left(K \leq 1.25\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \end{array} \]

Alternative 7
Error	26.1
Cost	7640

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U \leq -4.5 \cdot 10^{+177}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.2 \cdot 10^{+126}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{-2}{U} \cdot \left(J \cdot J\right) - U\\ \mathbf{elif}\;U \leq 5.5 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 5.9 \cdot 10^{+107}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.7 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{+207}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \frac{J}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Error	38.7
Cost	1368

\[\begin{array}{l} t_0 := \frac{-2}{U} \cdot \left(J \cdot J\right) - U\\ \mathbf{if}\;J \leq -1.1 \cdot 10^{+86}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.8 \cdot 10^{-82}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.5 \cdot 10^{-302}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{-276}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 155000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 9
Error	38.7
Cost	1368

\[\begin{array}{l} t_0 := \frac{-2}{U} \cdot \left(J \cdot J\right) - U\\ \mathbf{if}\;J \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.4 \cdot 10^{-78}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.6 \cdot 10^{-302}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-196}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 320000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 10
Error	38.6
Cost	984

\[\begin{array}{l} \mathbf{if}\;J \leq -1.35 \cdot 10^{+86}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -4.2 \cdot 10^{-80}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.35 \cdot 10^{-302}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.42 \cdot 10^{-272}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 420000:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 11
Error	47.0
Cost	656

\[\begin{array}{l} \mathbf{if}\;K \leq -8.4 \cdot 10^{+256}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -2.75 \cdot 10^{+169}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -1.2 \cdot 10^{-258}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 4.8 \cdot 10^{-225}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 12
Error	46.6
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023039 
(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)))))