Maksimov and Kolovsky, Equation (3)

Average Error: 17.6 → 8.4

Time: 17.2s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J \cdot -2\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\\ \mathbf{if}\;J \leq -1.4 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.7 \cdot 10^{-286}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* J (* 2.0 t_0))))))))
   (if (<= J -1.4e-211) t_1 (if (<= J -1.7e-286) 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 / (J * (2.0 * t_0)))));
	double tmp;
	if (J <= -1.4e-211) {
		tmp = t_1;
	} else if (J <= -1.7e-286) {
		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 / (J * (2.0 * t_0)))));
	double tmp;
	if (J <= -1.4e-211) {
		tmp = t_1;
	} else if (J <= -1.7e-286) {
		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 / (J * (2.0 * t_0)))))
	tmp = 0
	if J <= -1.4e-211:
		tmp = t_1
	elif J <= -1.7e-286:
		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(J * -2.0) * Float64(t_0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0))))))
	tmp = 0.0
	if (J <= -1.4e-211)
		tmp = t_1;
	elseif (J <= -1.7e-286)
		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 * -2.0) * (t_0 * hypot(1.0, (U / (J * (2.0 * t_0)))));
	tmp = 0.0;
	if (J <= -1.4e-211)
		tmp = t_1;
	elseif (J <= -1.7e-286)
		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[(J * -2.0), $MachinePrecision] * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.4e-211], t$95$1, If[LessEqual[J, -1.7e-286], U, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if J < -1.3999999999999999e-211 or -1.7000000000000001e-286 < J

   1. Initial program 16.1

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

   2. Simplified 6.8

      \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \]
      Proof
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (*.f64 J (*.f64 2 (cos.f64 (/.f64 K 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J 2) (cos.f64 (/.f64 K 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 J)) (cos.f64 (/.f64 K 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (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)))))))))): 42 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (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))))): 3 points increase in error, 8 points decrease in error

   if -1.3999999999999999e-211 < J < -1.7000000000000001e-286

   1. Initial program 41.2

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

   2. Simplified 24.8

      \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \]
      Proof
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (*.f64 J (*.f64 2 (cos.f64 (/.f64 K 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J 2) (cos.f64 (/.f64 K 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 J)) (cos.f64 (/.f64 K 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (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)))))))))): 42 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (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))))): 3 points increase in error, 8 points decrease in error

   3. Taylor expanded in U around -inf 33.2

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.4 \cdot 10^{-211}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq -1.7 \cdot 10^{-286}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.8
Cost	13960

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{if}\;U \leq -4.3 \cdot 10^{+140}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3.1 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot t_0\\ \mathbf{elif}\;U \leq 1.7 \cdot 10^{+239}:\\ \;\;\;\;J \cdot \left(-2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 2
Error	24.3
Cost	7568

\[\begin{array}{l} \mathbf{if}\;U \leq -5.2 \cdot 10^{+141}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.9 \cdot 10^{+73}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 4.8 \cdot 10^{-102}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;U \leq 8.2 \cdot 10^{+239}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 3
Error	26.7
Cost	7508

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U \leq -1.85 \cdot 10^{+140}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.3 \cdot 10^{+75}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 9.6 \cdot 10^{+69}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{+223}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 4
Error	38.7
Cost	1232

\[\begin{array}{l} \mathbf{if}\;U \leq -2.2 \cdot 10^{+141}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.55 \cdot 10^{+67}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 2.7 \cdot 10^{+155}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(0.5 \cdot \frac{U}{J} + -1\right)\\ \mathbf{elif}\;U \leq 5.5 \cdot 10^{+224}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5
Error	38.2
Cost	788

\[\begin{array}{l} \mathbf{if}\;U \leq -6.6 \cdot 10^{+141}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -3.3 \cdot 10^{+66}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 4.7 \cdot 10^{-17}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 4 \cdot 10^{+165}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.25 \cdot 10^{+223}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 6
Error	47.1
Cost	656

\[\begin{array}{l} \mathbf{if}\;U \leq -1.65 \cdot 10^{+140}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2 \cdot 10^{+162}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 8.5 \cdot 10^{+224}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7
Error	47.0
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022337 
(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)))))