Maksimov and Kolovsky, Equation (3)

Average Error: 17.8 → 7.9

Time: 17.7s

Precision: binary64

Cost: 20352

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right) \end{array} \]

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))))
   (* (* (* -2.0 J) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))))

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));
	return ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
}

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));
	return ((-2.0 * J) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
}

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))
	return ((-2.0 * J) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))

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))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))))
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 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
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]}, 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]]

Derivation

1. Initial program 17.8

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

2. Simplified 7.9

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

   [Start] 17.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   unpow2 [=>] 17.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
   hypot-1-def [=>] 7.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
   associate-*l* [=>] 7.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

3. Final simplification 7.9

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

Alternatives

Alternative 1
Error	20.2
Cost	27020

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t_0 \leq -0.45:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.254:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_0 \leq 0.88:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \end{array} \]

Alternative 2
Error	8.0
Cost	20352

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(-2 \cdot J\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right) \end{array} \]

Alternative 3
Error	16.4
Cost	13961

\[\begin{array}{l} \mathbf{if}\;J \leq -3.5 \cdot 10^{-250} \lor \neg \left(J \leq 1.2 \cdot 10^{-221}\right):\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \end{array} \]

Alternative 4
Error	26.5
Cost	7508

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.95 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4.7 \cdot 10^{-43}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.2 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{-250}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6 \cdot 10^{-127}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	38.7
Cost	1492

\[\begin{array}{l} \mathbf{if}\;J \leq -8.2 \cdot 10^{+69}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -3.8 \cdot 10^{-249}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.75 \cdot 10^{-68}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.02 \cdot 10^{+89}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{U}{J \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + -0.25 \cdot \frac{U}{J \cdot \frac{J}{U}}\right)\\ \end{array} \]

Alternative 6
Error	38.7
Cost	1236

\[\begin{array}{l} \mathbf{if}\;J \leq -4.1 \cdot 10^{+68}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.2 \cdot 10^{-250}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.3 \cdot 10^{-64}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.9 \cdot 10^{+37}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.35 \cdot 10^{+99}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{U}{J \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 7
Error	47.1
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -9.5 \cdot 10^{-58}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -3.2 \cdot 10^{-253}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -6.4 \cdot 10^{-289}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 2.4 \cdot 10^{-238}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 2.3 \cdot 10^{-184}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.1 \cdot 10^{+124}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8
Error	38.7
Cost	852

\[\begin{array}{l} \mathbf{if}\;J \leq -4.1 \cdot 10^{+68}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -3.4 \cdot 10^{-250}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-63}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.9 \cdot 10^{+37}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.6 \cdot 10^{+89}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 9
Error	47.0
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023073 
(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)))))