Maksimov and Kolovsky, Equation (3)

Average Accuracy: 72.2% → 86.5%

Time: 26.8s

Precision: binary64

Cost: 20617

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -2.6 \cdot 10^{-230} \lor \neg \left(J \leq 1.35 \cdot 10^{-269}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

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 -2.6e-230) (not (<= J 1.35e-269)))
     (* (* (* 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 <= -2.6e-230) || !(J <= 1.35e-269)) {
		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 <= -2.6e-230) || !(J <= 1.35e-269)) {
		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 <= -2.6e-230) or not (J <= 1.35e-269):
		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 <= -2.6e-230) || !(J <= 1.35e-269))
		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 <= -2.6e-230) || ~((J <= 1.35e-269)))
		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, -2.6e-230], N[Not[LessEqual[J, 1.35e-269]], $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 < -2.6000000000000001e-230 or 1.35000000000000008e-269 < J

   1. Initial program 76.8%

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

   2. Simplified 90.9%

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

      [Start] 76.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 76.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 90.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 90.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if -2.6000000000000001e-230 < J < 1.35000000000000008e-269

   1. Initial program 30.8%

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

   2. Simplified 53.2%

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

      [Start] 30.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 30.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 53.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 53.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   3. Taylor expanded in J around 0 45.8%

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

   4. Simplified 45.8%

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

      [Start] 45.8	\[ -1 \cdot U \]
      mul-1-neg [=>] 45.8	\[ \color{blue}{-U} \]

3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.6 \cdot 10^{-230} \lor \neg \left(J \leq 1.35 \cdot 10^{-269}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.2%
Cost	20617

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -2.6 \cdot 10^{-230} \lor \neg \left(J \leq 2.4 \cdot 10^{-257}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 2
Accuracy	72.8%
Cost	13960

\[\begin{array}{l} \mathbf{if}\;U \leq -1.95 \cdot 10^{+182}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.45 \cdot 10^{+174}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 3
Accuracy	68.0%
Cost	7569

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;K \leq -4 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -3.3 \cdot 10^{+94}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -290000000 \lor \neg \left(K \leq 0.24\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	58.1%
Cost	7508

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U \leq -5 \cdot 10^{+178}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -7.6 \cdot 10^{+45}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 4.4 \cdot 10^{+171}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5
Accuracy	26.6%
Cost	788

\[\begin{array}{l} \mathbf{if}\;K \leq -2.5 \cdot 10^{-14}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq -1.6 \cdot 10^{-206}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.15 \cdot 10^{-227}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 2.7 \cdot 10^{-90}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 7.2 \cdot 10^{+165}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 6
Accuracy	39.9%
Cost	788

\[\begin{array}{l} \mathbf{if}\;U \leq -1.95 \cdot 10^{+177}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.85 \cdot 10^{+36}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.8 \cdot 10^{+19}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{+50}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3.5 \cdot 10^{+173}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7
Accuracy	26.9%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023129 
(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)))))