Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 9.1

Time: 22.3s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right) \cdot \left(t_0 \cdot -2\right)\right)\\ \mathbf{if}\;J \leq -3.7 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-219}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* (hypot 1.0 (/ U (* t_0 (* J 2.0)))) (* t_0 -2.0)))))
   (if (<= J -3.7e-240) t_1 (if (<= J 2e-219) 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 * (hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0));
	double tmp;
	if (J <= -3.7e-240) {
		tmp = t_1;
	} else if (J <= 2e-219) {
		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 * (Math.hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0));
	double tmp;
	if (J <= -3.7e-240) {
		tmp = t_1;
	} else if (J <= 2e-219) {
		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 * (math.hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0))
	tmp = 0
	if J <= -3.7e-240:
		tmp = t_1
	elif J <= 2e-219:
		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(J * Float64(hypot(1.0, Float64(U / Float64(t_0 * Float64(J * 2.0)))) * Float64(t_0 * -2.0)))
	tmp = 0.0
	if (J <= -3.7e-240)
		tmp = t_1;
	elseif (J <= 2e-219)
		tmp = 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 * (hypot(1.0, (U / (t_0 * (J * 2.0)))) * (t_0 * -2.0));
	tmp = 0.0;
	if (J <= -3.7e-240)
		tmp = t_1;
	elseif (J <= 2e-219)
		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[(J * N[(N[Sqrt[1.0 ^ 2 + N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -3.7e-240], t$95$1, If[LessEqual[J, 2e-219], U, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if J < -3.7000000000000002e-240 or 2.0000000000000001e-219 < J

   1. Initial program 14.2

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

   2. Simplified 5.2

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      Proof
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (*.f64 J 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (Rewrite<= *-commutative_binary64 (*.f64 2 J))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 37 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 -2 (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J -2) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 J)) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 2 points increase in error, 8 points decrease in error

   if -3.7000000000000002e-240 < J < 2.0000000000000001e-219

   1. Initial program 43.4

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

   2. Taylor expanded in U around -inf 35.3

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.7 \cdot 10^{-240}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-219}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.4
Cost	14092

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ t_1 := J \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot -2\right) \cdot t_0\right)\\ \mathbf{if}\;J \leq -3.5 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{-233}:\\ \;\;\;\;J \cdot \left(-2 \cdot t_0\right)\\ \mathbf{elif}\;J \leq 2.65 \cdot 10^{-219}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	25.9
Cost	7376

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -6 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.1 \cdot 10^{-183}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.4 \cdot 10^{-220}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6 \cdot 10^{-131}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.2
Cost	7304

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;K \leq -2.6 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq 6.6 \cdot 10^{+30}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	38.8
Cost	1760

\[\begin{array}{l} t_0 := U \cdot \left(\frac{U}{J} \cdot -0.25\right) + J \cdot -2\\ \mathbf{if}\;J \leq -2.8 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -8.6 \cdot 10^{+30}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -2.15 \cdot 10^{-20}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -1.35 \cdot 10^{-183}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.8 \cdot 10^{-220}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.9 \cdot 10^{-232}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-12}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 6.5 \cdot 10^{+22}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	38.6
Cost	1248

\[\begin{array}{l} \mathbf{if}\;J \leq -8.2 \cdot 10^{+77}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{+30}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1.6 \cdot 10^{-21}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -4.5 \cdot 10^{-184}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.2 \cdot 10^{-220}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{-232}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 58000000000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 6
Error	47.3
Cost	920

\[\begin{array}{l} \mathbf{if}\;K \leq -2.4 \cdot 10^{+107}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.8 \cdot 10^{-243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 3.7 \cdot 10^{-262}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.55 \cdot 10^{-207}:\\ \;\;\;\;-U\\ \mathbf{elif}\;K \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 4.1 \cdot 10^{+142}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 7
Error	47.3
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022318 
(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)))))