Maksimov and Kolovsky, Equation (3)

Average Error: 17.9 → 8.7

Time: 23.9s

Precision: binary64

Cost: 20617

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -1.25 \cdot 10^{-187} \lor \neg \left(J \leq -4.4 \cdot 10^{-243}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

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 -1.25e-187) (not (<= J -4.4e-243)))
     (* (* (* 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 <= -1.25e-187) || !(J <= -4.4e-243)) {
		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 <= -1.25e-187) || !(J <= -4.4e-243)) {
		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 <= -1.25e-187) or not (J <= -4.4e-243):
		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 <= -1.25e-187) || !(J <= -4.4e-243))
		tmp = Float64(Float64(Float64(J * -2.0) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	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 ((J <= -1.25e-187) || ~((J <= -4.4e-243)))
		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, -1.25e-187], N[Not[LessEqual[J, -4.4e-243]], $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 < -1.2499999999999999e-187 or -4.3999999999999998e-243 < J

   1. Initial program 17.1

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

   2. Simplified 7.6

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

      [Start] 17.1	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 17.1	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 7.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 7.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if -1.2499999999999999e-187 < J < -4.3999999999999998e-243

   1. Initial program 36.3

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

   2. Simplified 23.5

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

      [Start] 36.3	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 36.4	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 36.4	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 23.5	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 23.5	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 23.5	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in J around 0 33.8

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

   4. Simplified 33.8

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

      [Start] 33.8	\[ -1 \cdot U \]
      mul-1-neg [=>] 33.8	\[ \color{blue}{-U} \]

3. Recombined 2 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.25 \cdot 10^{-187} \lor \neg \left(J \leq -4.4 \cdot 10^{-243}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternatives

Alternative 1
Error	8.7
Cost	20617

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -7 \cdot 10^{-198} \lor \neg \left(J \leq -7.2 \cdot 10^{-241}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 2
Error	26.4
Cost	14557

\[\begin{array}{l} t_0 := U + -2 \cdot \left(J \cdot -2\right)\\ t_1 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -8.1 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.08 \cdot 10^{-240}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -6 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 2.05 \cdot 10^{-261}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{+25} \lor \neg \left(J \leq 7.4 \cdot 10^{+118}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U}{\frac{J}{U}}}{J}, 1\right)}\\ \end{array} \]

Alternative 3
Error	16.5
Cost	14224

\[\begin{array}{l} t_0 := \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{if}\;J \leq -5.7 \cdot 10^{-188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.15 \cdot 10^{-240}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -3.35 \cdot 10^{-292}:\\ \;\;\;\;U + -2 \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;J \leq 8 \cdot 10^{-253}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	25.9
Cost	7244

\[\begin{array}{l} \mathbf{if}\;U \leq -1.05 \cdot 10^{+178}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.45 \cdot 10^{+89}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{+59}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{+184}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5
Error	38.1
Cost	1248

\[\begin{array}{l} \mathbf{if}\;J \leq -2.3 \cdot 10^{-95}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -3.25 \cdot 10^{-257}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1.1 \cdot 10^{-292}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 10^{-136}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6.5 \cdot 10^{-22}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq 1.35 \cdot 10^{+58}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 6
Error	38.0
Cost	1248

\[\begin{array}{l} \mathbf{if}\;J \leq -1.25 \cdot 10^{-95}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -1.08 \cdot 10^{-240}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -3.6 \cdot 10^{-291}:\\ \;\;\;\;U + -2 \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;J \leq 10^{-260}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{-137}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6.4 \cdot 10^{-22}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq 5.7 \cdot 10^{+57}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 7
Error	47.0
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -2.8 \cdot 10^{+80}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.2 \cdot 10^{-120}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -7.2 \cdot 10^{-273}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.25 \cdot 10^{-172}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 8 \cdot 10^{-90}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 2.7 \cdot 10^{+147}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	46.9
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023060 
(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)))))