Maksimov and Kolovsky, Equation (3)

Average Error: 18.7 → 9.5

Time: 16.3s

Precision: binary64

Cost: 20880

Math

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

↓

\[\begin{array}{l} t_0 := J \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_0}\right)\right)\\ \mathbf{if}\;J \leq -8.414332066908579 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 4.4902067410718914 \cdot 10^{-296}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 2.0454703649906397 \cdot 10^{-219}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* J (cos (/ K 2.0))))
        (t_1 (* -2.0 (* t_0 (hypot 1.0 (/ U (* 2.0 t_0)))))))
   (if (<= J -8.414332066908579e-128)
     t_1
     (if (<= J -7.075848735380914e-175)
       (* -2.0 (* U -0.5))
       (if (<= J 4.4902067410718914e-296)
         t_1
         (if (<= J 2.0454703649906397e-219) (* -2.0 (* U 0.5)) 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 = J * cos((K / 2.0));
	double t_1 = -2.0 * (t_0 * hypot(1.0, (U / (2.0 * t_0))));
	double tmp;
	if (J <= -8.414332066908579e-128) {
		tmp = t_1;
	} else if (J <= -7.075848735380914e-175) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 4.4902067410718914e-296) {
		tmp = t_1;
	} else if (J <= 2.0454703649906397e-219) {
		tmp = -2.0 * (U * 0.5);
	} 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 = J * Math.cos((K / 2.0));
	double t_1 = -2.0 * (t_0 * Math.hypot(1.0, (U / (2.0 * t_0))));
	double tmp;
	if (J <= -8.414332066908579e-128) {
		tmp = t_1;
	} else if (J <= -7.075848735380914e-175) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 4.4902067410718914e-296) {
		tmp = t_1;
	} else if (J <= 2.0454703649906397e-219) {
		tmp = -2.0 * (U * 0.5);
	} 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 = J * math.cos((K / 2.0))
	t_1 = -2.0 * (t_0 * math.hypot(1.0, (U / (2.0 * t_0))))
	tmp = 0
	if J <= -8.414332066908579e-128:
		tmp = t_1
	elif J <= -7.075848735380914e-175:
		tmp = -2.0 * (U * -0.5)
	elif J <= 4.4902067410718914e-296:
		tmp = t_1
	elif J <= 2.0454703649906397e-219:
		tmp = -2.0 * (U * 0.5)
	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 = Float64(J * cos(Float64(K / 2.0)))
	t_1 = Float64(-2.0 * Float64(t_0 * hypot(1.0, Float64(U / Float64(2.0 * t_0)))))
	tmp = 0.0
	if (J <= -8.414332066908579e-128)
		tmp = t_1;
	elseif (J <= -7.075848735380914e-175)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 4.4902067410718914e-296)
		tmp = t_1;
	elseif (J <= 2.0454703649906397e-219)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	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 = J * cos((K / 2.0));
	t_1 = -2.0 * (t_0 * hypot(1.0, (U / (2.0 * t_0))));
	tmp = 0.0;
	if (J <= -8.414332066908579e-128)
		tmp = t_1;
	elseif (J <= -7.075848735380914e-175)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 4.4902067410718914e-296)
		tmp = t_1;
	elseif (J <= 2.0454703649906397e-219)
		tmp = -2.0 * (U * 0.5);
	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[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -8.414332066908579e-128], t$95$1, If[LessEqual[J, -7.075848735380914e-175], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.4902067410718914e-296], t$95$1, If[LessEqual[J, 2.0454703649906397e-219], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -8.4143320669085788e-128 or -7.07584873538091396e-175 < J < 4.4902067410718914e-296 or 2.04547036499063975e-219 < J

   1. Initial program 16.2

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

   2. Simplified 6.3

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

   if -8.4143320669085788e-128 < J < -7.07584873538091396e-175

   1. Initial program 34.7

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

   2. Simplified 15.1

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

   3. Taylor expanded in U around -inf 39.6

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]

   4. Simplified 39.6

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5\right)} \]

   if 4.4902067410718914e-296 < J < 2.04547036499063975e-219

   1. Initial program 43.0

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

   2. Simplified 27.0

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

   3. Applied egg-rr 27.8

      \[\leadsto -2 \cdot \color{blue}{{\left(\sqrt[3]{J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)}\right)\right)}\right)}^{3}} \]

   4. Applied egg-rr 27.9

      \[\leadsto -2 \cdot {\color{blue}{\left(\sqrt[3]{J} \cdot \sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(K \cdot 0.5\right) \cdot \left(J + J\right)}\right)}\right)}}^{3} \]

   5. Taylor expanded in J around 0 35.2

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot \left({1}^{0.3333333333333333} \cdot U\right)\right)} \]

   6. Simplified 35.2

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8.414332066908579 \cdot 10^{-128}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 4.4902067410718914 \cdot 10^{-296}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq 2.0454703649906397 \cdot 10^{-219}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.4
Cost	14356

\[\begin{array}{l} t_0 := -2 \cdot \left(U \cdot 0.5\right)\\ t_1 := \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ t_2 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot t_1\right)\\ \mathbf{if}\;J \leq -8.414332066908579 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -1.967268705511324 \cdot 10^{-286}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 4.4902067410718914 \cdot 10^{-296}:\\ \;\;\;\;-2 \cdot \left(J \cdot t_1\right)\\ \mathbf{elif}\;J \leq 4.757294762352974 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	18.4
Cost	14356

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ t_1 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot t_0\right)\\ \mathbf{if}\;J \leq -8.414332066908579 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -1.967268705511324 \cdot 10^{-286}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, \frac{J \cdot \left(0.5 + 0.5 \cdot \cos K\right)}{\frac{U}{J}}\right)\\ \mathbf{elif}\;J \leq 4.4902067410718914 \cdot 10^{-296}:\\ \;\;\;\;-2 \cdot \left(J \cdot t_0\right)\\ \mathbf{elif}\;J \leq 4.757294762352974 \cdot 10^{-150}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	25.3
Cost	8096

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ t_1 := -2 \cdot \left(U \cdot 0.5\right)\\ t_2 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -5.239959057112384 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq -14671848055125152:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2.5204116505435214 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq -1.967268705511324 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 4.4902067410718914 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 2.0454703649906397 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 5.66951041232629 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	26.5
Cost	7372

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -2.5204116505435214 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 2.8465762312540636 \cdot 10^{-112}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, J \cdot \frac{J}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	26.5
Cost	7244

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -2.5204116505435214 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 2.8465762312540636 \cdot 10^{-112}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	38.3
Cost	980

\[\begin{array}{l} t_0 := -2 \cdot \left(U \cdot 0.5\right)\\ t_1 := -2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{if}\;J \leq -6.447339204477278 \cdot 10^{-16}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -7.075848735380914 \cdot 10^{-175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 5.030956872729535 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 1.1161106240058251 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 4.276426523173706:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 7
Error	38.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;J \leq -1.4801396040192268 \cdot 10^{-19}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq 6.49243747671959 \cdot 10^{-94}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 8
Error	45.6
Cost	192

\[J \cdot -2 \]

Reproduce

herbie shell --seed 2022225 
(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)))))