Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 11.1

Time: 16.8s

Precision: binary64

Cost: 20880

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{if}\;J \leq -8.2 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{-290}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-195}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 8.8 \cdot 10^{-178}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* (* (* J -2.0) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))))
   (if (<= J -8.2e-134)
     t_1
     (if (<= J 2.4e-290)
       (- U)
       (if (<= J 1.25e-195) U (if (<= J 8.8e-178) (- U) t_1))))))

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 = ((J * -2.0) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	double tmp;
	if (J <= -8.2e-134) {
		tmp = t_1;
	} else if (J <= 2.4e-290) {
		tmp = -U;
	} else if (J <= 1.25e-195) {
		tmp = U;
	} else if (J <= 8.8e-178) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	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 = ((J * -2.0) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	double tmp;
	if (J <= -8.2e-134) {
		tmp = t_1;
	} else if (J <= 2.4e-290) {
		tmp = -U;
	} else if (J <= 1.25e-195) {
		tmp = U;
	} else if (J <= 8.8e-178) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	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 = ((J * -2.0) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	tmp = 0
	if J <= -8.2e-134:
		tmp = t_1
	elif J <= 2.4e-290:
		tmp = -U
	elif J <= 1.25e-195:
		tmp = U
	elif J <= 8.8e-178:
		tmp = -U
	else:
		tmp = t_1
	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(Float64(J * -2.0) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))))
	tmp = 0.0
	if (J <= -8.2e-134)
		tmp = t_1;
	elseif (J <= 2.4e-290)
		tmp = Float64(-U);
	elseif (J <= 1.25e-195)
		tmp = U;
	elseif (J <= 8.8e-178)
		tmp = Float64(-U);
	else
		tmp = t_1;
	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 = ((J * -2.0) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	tmp = 0.0;
	if (J <= -8.2e-134)
		tmp = t_1;
	elseif (J <= 2.4e-290)
		tmp = -U;
	elseif (J <= 1.25e-195)
		tmp = U;
	elseif (J <= 8.8e-178)
		tmp = -U;
	else
		tmp = t_1;
	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[(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]}, If[LessEqual[J, -8.2e-134], t$95$1, If[LessEqual[J, 2.4e-290], (-U), If[LessEqual[J, 1.25e-195], U, If[LessEqual[J, 8.8e-178], (-U), t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -8.2000000000000004e-134 or 8.8000000000000005e-178 < J

   1. Initial program 11.3

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

   2. Simplified 2.9

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

      [Start] 11.3	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 11.3	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 2.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 2.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if -8.2000000000000004e-134 < J < 2.4000000000000001e-290 or 1.25000000000000002e-195 < J < 8.8000000000000005e-178

   1. Initial program 38.9

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

   2. Simplified 22.6

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

      [Start] 38.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 38.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 22.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 22.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Taylor expanded in J around 0 35.2

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

   4. Simplified 35.2

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

      [Start] 35.2	\[ -1 \cdot U \]
      mul-1-neg [=>] 35.2	\[ \color{blue}{-U} \]

   if 2.4000000000000001e-290 < J < 1.25000000000000002e-195

   1. Initial program 40.2

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

   2. Simplified 22.6

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

      [Start] 40.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 40.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 22.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 22.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Taylor expanded in U around -inf 36.3

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8.2 \cdot 10^{-134}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{-290}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.25 \cdot 10^{-195}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 8.8 \cdot 10^{-178}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	23.7
Cost	7832

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ t_1 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.4 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 2.5 \cdot 10^{-290}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.06 \cdot 10^{-197}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{-176}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	26.2
Cost	7376

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -6.4 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 7.5 \cdot 10^{-283}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.3 \cdot 10^{-120}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.35 \cdot 10^{-62}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	46.7
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -8.6 \cdot 10^{+54}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -0.9:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -7.5 \cdot 10^{-159}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 3 \cdot 10^{-174}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 2.75 \cdot 10^{+240}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 4
Error	38.4
Cost	788

\[\begin{array}{l} \mathbf{if}\;U \leq -8.4 \cdot 10^{+33}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 7.2 \cdot 10^{+198}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.5 \cdot 10^{+214}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5
Error	46.8
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023046 
(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)))))