Average Error: 17.8 → 8.6
Time: 17.5s
Precision: binary64
Cost: 20616
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
↓
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right) \cdot \left(t_0 \cdot -2\right)\right)\\ \mathbf{if}\;J \leq -2.1141970306468305 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 1.6533316064809362 \cdot 10^{-249}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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 (<= J -2.1141970306468305e-257)
     t_1
     (if (<= J 1.6533316064809362e-249) (- 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 (J <= -2.1141970306468305e-257) {
		tmp = t_1;
	} else if (J <= 1.6533316064809362e-249) {
		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 (J <= -2.1141970306468305e-257) {
		tmp = t_1;
	} else if (J <= 1.6533316064809362e-249) {
		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 J <= -2.1141970306468305e-257:
		tmp = t_1
	elif J <= 1.6533316064809362e-249:
		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 (J <= -2.1141970306468305e-257)
		tmp = t_1;
	elseif (J <= 1.6533316064809362e-249)
		tmp = Float64(-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 (J <= -2.1141970306468305e-257)
		tmp = t_1;
	elseif (J <= 1.6533316064809362e-249)
		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[J, -2.1141970306468305e-257], t$95$1, If[LessEqual[J, 1.6533316064809362e-249], (-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}\;J \leq -2.1141970306468305 \cdot 10^{-257}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;J \leq 1.6533316064809362 \cdot 10^{-249}:\\
\;\;\;\;-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 J < -2.1141970306468305e-257 or 1.65333160648093621e-249 < J

    1. Initial program 15.2

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

    2. Simplified6.2

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      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))))): 38 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, 9 points decrease in error

    if -2.1141970306468305e-257 < J < 1.65333160648093621e-249

    1. Initial program 43.9

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

    2. Simplified27.0

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      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))))): 38 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, 9 points decrease in error

    3. Taylor expanded in J around 0 33.7

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

    4. Simplified33.7

      \[\leadsto \color{blue}{-U} \]
      Proof
      (neg.f64 U): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 U)): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.1141970306468305 \cdot 10^{-257}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;J \leq 1.6533316064809362 \cdot 10^{-249}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error16.7
Cost13960
\[\begin{array}{l} \mathbf{if}\;U \leq -3.4973344666273984 \cdot 10^{+213}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.0937443210938752 \cdot 10^{+228}:\\ \;\;\;\;J \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot -2\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
Alternative 2
Error26.1
Cost7244
\[\begin{array}{l} \mathbf{if}\;U \leq -3.4973344666273984 \cdot 10^{+213}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -3.176469001613169 \cdot 10^{+19}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 6.3905968945692675 \cdot 10^{+162}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
Alternative 3
Error37.8
Cost788
\[\begin{array}{l} \mathbf{if}\;U \leq -3.4973344666273984 \cdot 10^{+213}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -16608.73371968515:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.067988339048737 \cdot 10^{+48}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 2.5157653156030647 \cdot 10^{+119}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 6.3905968945692675 \cdot 10^{+162}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
Alternative 4
Error47.4
Cost524
\[\begin{array}{l} \mathbf{if}\;U \leq -3.4973344666273984 \cdot 10^{+213}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.8529782156145774 \cdot 10^{-230}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 6.3905968945692675 \cdot 10^{+162}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
Alternative 5
Error47.2
Cost64
\[U \]

Error

Reproduce

herbie shell --seed 2022293 
(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)))))