Average Error: 18.0 → 9.2
Time: 21.0s
Precision: binary64
Cost: 20748
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right) \cdot \left(t_0 \cdot -2\right)\right)\\ \mathbf{if}\;U \leq -2.1046685566989484 \cdot 10^{+186}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.4376729941037334 \cdot 10^{+247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.822574863810491 \cdot 10^{+286}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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 (<= U -2.1046685566989484e+186)
     U
     (if (<= U 3.4376729941037334e+247)
       t_1
       (if (<= U 1.822574863810491e+286) U t_1)))))
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 (U <= -2.1046685566989484e+186) {
		tmp = U;
	} else if (U <= 3.4376729941037334e+247) {
		tmp = t_1;
	} else if (U <= 1.822574863810491e+286) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	return tmp;
}
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 (U <= -2.1046685566989484e+186) {
		tmp = U;
	} else if (U <= 3.4376729941037334e+247) {
		tmp = t_1;
	} else if (U <= 1.822574863810491e+286) {
		tmp = U;
	} else {
		tmp = t_1;
	}
	return tmp;
}
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 U <= -2.1046685566989484e+186:
		tmp = U
	elif U <= 3.4376729941037334e+247:
		tmp = t_1
	elif U <= 1.822574863810491e+286:
		tmp = U
	else:
		tmp = t_1
	return tmp
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 (U <= -2.1046685566989484e+186)
		tmp = U;
	elseif (U <= 3.4376729941037334e+247)
		tmp = t_1;
	elseif (U <= 1.822574863810491e+286)
		tmp = U;
	else
		tmp = t_1;
	end
	return tmp
end
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 (U <= -2.1046685566989484e+186)
		tmp = U;
	elseif (U <= 3.4376729941037334e+247)
		tmp = t_1;
	elseif (U <= 1.822574863810491e+286)
		tmp = U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end
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[U, -2.1046685566989484e+186], U, If[LessEqual[U, 3.4376729941037334e+247], t$95$1, If[LessEqual[U, 1.822574863810491e+286], U, t$95$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}}
\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}\;U \leq -2.1046685566989484 \cdot 10^{+186}:\\
\;\;\;\;U\\

\mathbf{elif}\;U \leq 3.4376729941037334 \cdot 10^{+247}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;U \leq 1.822574863810491 \cdot 10^{+286}:\\
\;\;\;\;U\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if U < -2.1046685566989484e186 or 3.43767299410373345e247 < U < 1.8225748638104909e286

    1. Initial program 42.5

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
    2. Simplified26.6

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      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))))): 32 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, 18 points decrease in error
    3. Taylor expanded in U around -inf 47.4

      \[\leadsto J \cdot \color{blue}{\left(2 \cdot \frac{{\cos \left(0.5 \cdot K\right)}^{2} \cdot J}{U} + \frac{U}{J}\right)} \]
    4. Taylor expanded in J around 0 34.9

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

    if -2.1046685566989484e186 < U < 3.43767299410373345e247 or 1.8225748638104909e286 < U

    1. Initial program 14.4

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
    2. Simplified5.4

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      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))))): 32 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, 18 points decrease in error
  3. Recombined 2 regimes into one program.
  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.1046685566989484 \cdot 10^{+186}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.4376729941037334 \cdot 10^{+247}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;U \leq 1.822574863810491 \cdot 10^{+286}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error27.4
Cost14292
\[\begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;K \leq -1.587743219962244 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -9.765522932529785 \cdot 10^{+58}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -2.8760891655345143 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -31223455610127570:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq 1.4516224170385698 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U}{J} \cdot \frac{U}{J}, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;K \leq 7.524274976103549 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq 1.8047380258551667 \cdot 10^{+275}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error25.7
Cost7112
\[\begin{array}{l} \mathbf{if}\;U \leq -1.8514338850953588 \cdot 10^{+78}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.295297684202953 \cdot 10^{+59}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
Alternative 3
Error38.3
Cost836
\[\begin{array}{l} \mathbf{if}\;J \leq -2.9292104967402845 \cdot 10^{+83}:\\ \;\;\;\;-0.25 \cdot \left(U \cdot \frac{U}{J}\right) + J \cdot -2\\ \mathbf{elif}\;J \leq -89798680573.09439:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -6.057669353767584 \cdot 10^{-107}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq 8.448359197294565 \cdot 10^{+31}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]
Alternative 4
Error38.3
Cost720
\[\begin{array}{l} \mathbf{if}\;J \leq -2.9292104967402845 \cdot 10^{+83}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -89798680573.09439:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -6.057669353767584 \cdot 10^{-107}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq 8.448359197294565 \cdot 10^{+31}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]
Alternative 5
Error46.8
Cost392
\[\begin{array}{l} \mathbf{if}\;U \leq -1.8206504013017795 \cdot 10^{-20}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.0762433267512488 \cdot 10^{-159}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
Alternative 6
Error46.8
Cost64
\[U \]

Error

Reproduce

herbie shell --seed 2022308 
(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)))))