Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.0% → 99.4%
Time: 13.3s
Alternatives: 6
Speedup: 1.9×

Specification

?
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \cos \left(\frac{K}{2}\right)\\
\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}}
\end{array}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 6 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 72.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \cos \left(\frac{K}{2}\right)\\
\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}}
\end{array}
\end{array}

Alternative 1: 99.4% accurate, 0.4× speedup?

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J_m \cdot t_0\\ t_2 := \left(\left(-2 \cdot J_m\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U_m}{t_0 \cdot \left(J_m \cdot 2\right)}\right)}^{2}}\\ J_s \cdot \begin{array}{l} \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot -0.5\right)\\ \end{array} \end{array} \end{array} \]
U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* J_m t_0))
        (t_2
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* -2.0 (* U_m 0.5))
      (if (<= t_2 5e+296)
        (* -2.0 (* t_1 (hypot 1.0 (/ (/ U_m 2.0) t_1))))
        (* -2.0 (* U_m -0.5)))))))
U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = J_m * t_0;
	double t_2 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (U_m * 0.5);
	} else if (t_2 <= 5e+296) {
		tmp = -2.0 * (t_1 * hypot(1.0, ((U_m / 2.0) / t_1)));
	} else {
		tmp = -2.0 * (U_m * -0.5);
	}
	return J_s * tmp;
}

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = J_m * t_0;
	double t_2 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (U_m * 0.5);
	} else if (t_2 <= 5e+296) {
		tmp = -2.0 * (t_1 * Math.hypot(1.0, ((U_m / 2.0) / t_1)));
	} else {
		tmp = -2.0 * (U_m * -0.5);
	}
	return J_s * tmp;
}

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = J_m * t_0
	t_2 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (U_m * 0.5)
	elif t_2 <= 5e+296:
		tmp = -2.0 * (t_1 * math.hypot(1.0, ((U_m / 2.0) / t_1)))
	else:
		tmp = -2.0 * (U_m * -0.5)
	return J_s * tmp

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J_m * t_0)
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	elseif (t_2 <= 5e+296)
		tmp = Float64(-2.0 * Float64(t_1 * hypot(1.0, Float64(Float64(U_m / 2.0) / t_1))));
	else
		tmp = Float64(-2.0 * Float64(U_m * -0.5));
	end
	return Float64(J_s * tmp)
end

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = J_m * t_0;
	t_2 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J_m * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (U_m * 0.5);
	elseif (t_2 <= 5e+296)
		tmp = -2.0 * (t_1 * hypot(1.0, ((U_m / 2.0) / t_1)));
	else
		tmp = -2.0 * (U_m * -0.5);
	end
	tmp_2 = J_s * tmp;
end

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(J$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+296], N[(-2.0 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U$95$m * -0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

\begin{array}{l}
U_m = \left|U\right|
\\
J_m = \left|J\right|
\\
J_s = \mathsf{copysign}\left(1, J\right)

\\
\begin{array}{l}
t_0 := \cos \left(\frac{K}{2}\right)\\
t_1 := J_m \cdot t_0\\
t_2 := \left(\left(-2 \cdot J_m\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U_m}{t_0 \cdot \left(J_m \cdot 2\right)}\right)}^{2}}\\
J_s \cdot \begin{array}{l}
\mathbf{if}\;t_2 \leq -\infty:\\
\;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\

\mathbf{elif}\;t_2 \leq 5 \cdot 10^{+296}:\\
\;\;\;\;-2 \cdot \left(t_1 \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{t_1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(U_m \cdot -0.5\right)\\


\end{array}
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 89.2% accurate, 1.3× speedup?

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J_s \cdot \begin{array}{l} \mathbf{if}\;J_m \leq 6.6 \cdot 10^{-220}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J_m \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{J_m \cdot t_0}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]
U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<= J_m 6.6e-220)
      (* -2.0 (* U_m 0.5))
      (* -2.0 (* t_0 (* J_m (hypot 1.0 (/ (/ U_m 2.0) (* J_m t_0))))))))))
U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (J_m <= 6.6e-220) {
		tmp = -2.0 * (U_m * 0.5);
	} else {
		tmp = -2.0 * (t_0 * (J_m * hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))));
	}
	return J_s * tmp;
}

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (J_m <= 6.6e-220) {
		tmp = -2.0 * (U_m * 0.5);
	} else {
		tmp = -2.0 * (t_0 * (J_m * Math.hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))));
	}
	return J_s * tmp;
}

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if J_m <= 6.6e-220:
		tmp = -2.0 * (U_m * 0.5)
	else:
		tmp = -2.0 * (t_0 * (J_m * math.hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))))
	return J_s * tmp

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (J_m <= 6.6e-220)
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	else
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J_m * hypot(1.0, Float64(Float64(U_m / 2.0) / Float64(J_m * t_0))))));
	end
	return Float64(J_s * tmp)
end

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (J_m <= 6.6e-220)
		tmp = -2.0 * (U_m * 0.5);
	else
		tmp = -2.0 * (t_0 * (J_m * hypot(1.0, ((U_m / 2.0) / (J_m * t_0)))));
	end
	tmp_2 = J_s * tmp;
end

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[J$95$m, 6.6e-220], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$0 * N[(J$95$m * N[Sqrt[1.0 ^ 2 + N[(N[(U$95$m / 2.0), $MachinePrecision] / N[(J$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

\begin{array}{l}
U_m = \left|U\right|
\\
J_m = \left|J\right|
\\
J_s = \mathsf{copysign}\left(1, J\right)

\\
\begin{array}{l}
t_0 := \cos \left(\frac{K}{2}\right)\\
J_s \cdot \begin{array}{l}
\mathbf{if}\;J_m \leq 6.6 \cdot 10^{-220}:\\
\;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(t_0 \cdot \left(J_m \cdot \mathsf{hypot}\left(1, \frac{\frac{U_m}{2}}{J_m \cdot t_0}\right)\right)\right)\\


\end{array}
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 77.7% accurate, 1.9× speedup?

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J_s \cdot \begin{array}{l} \mathbf{if}\;U_m \leq 1.5 \cdot 10^{+195}:\\ \;\;\;\;-2 \cdot \left(\left(J_m \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U_m \cdot \frac{0.5}{J_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \end{array} \end{array} \]
U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1.5e+195)
    (* -2.0 (* (* J_m (cos (/ K 2.0))) (hypot 1.0 (* U_m (/ 0.5 J_m)))))
    (* -2.0 (* U_m 0.5)))))
U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.5e+195) {
		tmp = -2.0 * ((J_m * cos((K / 2.0))) * hypot(1.0, (U_m * (0.5 / J_m))));
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return J_s * tmp;
}

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.5e+195) {
		tmp = -2.0 * ((J_m * Math.cos((K / 2.0))) * Math.hypot(1.0, (U_m * (0.5 / J_m))));
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return J_s * tmp;
}

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1.5e+195:
		tmp = -2.0 * ((J_m * math.cos((K / 2.0))) * math.hypot(1.0, (U_m * (0.5 / J_m))))
	else:
		tmp = -2.0 * (U_m * 0.5)
	return J_s * tmp

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.5e+195)
		tmp = Float64(-2.0 * Float64(Float64(J_m * cos(Float64(K / 2.0))) * hypot(1.0, Float64(U_m * Float64(0.5 / J_m)))));
	else
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	end
	return Float64(J_s * tmp)
end

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.5e+195)
		tmp = -2.0 * ((J_m * cos((K / 2.0))) * hypot(1.0, (U_m * (0.5 / J_m))));
	else
		tmp = -2.0 * (U_m * 0.5);
	end
	tmp_2 = J_s * tmp;
end

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.5e+195], N[(-2.0 * N[(N[(J$95$m * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U$95$m * N[(0.5 / J$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

\begin{array}{l}
U_m = \left|U\right|
\\
J_m = \left|J\right|
\\
J_s = \mathsf{copysign}\left(1, J\right)

\\
J_s \cdot \begin{array}{l}
\mathbf{if}\;U_m \leq 1.5 \cdot 10^{+195}:\\
\;\;\;\;-2 \cdot \left(\left(J_m \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, U_m \cdot \frac{0.5}{J_m}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 66.9% accurate, 3.9× speedup?

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J_s \cdot \begin{array}{l} \mathbf{if}\;U_m \leq 1300000000:\\ \;\;\;\;-2 \cdot \left(J_m \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \end{array} \end{array} \]
U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1300000000.0)
    (* -2.0 (* J_m (cos (/ K 2.0))))
    (* -2.0 (* U_m 0.5)))))
U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1300000000.0) {
		tmp = -2.0 * (J_m * cos((K / 2.0)));
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return J_s * tmp;
}

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1300000000.0d0) then
        tmp = (-2.0d0) * (j_m * cos((k / 2.0d0)))
    else
        tmp = (-2.0d0) * (u_m * 0.5d0)
    end if
    code = j_s * tmp
end function

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1300000000.0) {
		tmp = -2.0 * (J_m * Math.cos((K / 2.0)));
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return J_s * tmp;
}

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if U_m <= 1300000000.0:
		tmp = -2.0 * (J_m * math.cos((K / 2.0)))
	else:
		tmp = -2.0 * (U_m * 0.5)
	return J_s * tmp

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1300000000.0)
		tmp = Float64(-2.0 * Float64(J_m * cos(Float64(K / 2.0))));
	else
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	end
	return Float64(J_s * tmp)
end

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (U_m <= 1300000000.0)
		tmp = -2.0 * (J_m * cos((K / 2.0)));
	else
		tmp = -2.0 * (U_m * 0.5);
	end
	tmp_2 = J_s * tmp;
end

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1300000000.0], N[(-2.0 * N[(J$95$m * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

\begin{array}{l}
U_m = \left|U\right|
\\
J_m = \left|J\right|
\\
J_s = \mathsf{copysign}\left(1, J\right)

\\
J_s \cdot \begin{array}{l}
\mathbf{if}\;U_m \leq 1300000000:\\
\;\;\;\;-2 \cdot \left(J_m \cdot \cos \left(\frac{K}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 50.6% accurate, 37.5× speedup?

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J_s \cdot \begin{array}{l} \mathbf{if}\;U_m \leq 1.75 \cdot 10^{-71} \lor \neg \left(U_m \leq 1.9 \cdot 10^{-29}\right) \land U_m \leq 0.72:\\ \;\;\;\;-2 \cdot J_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\ \end{array} \end{array} \]
U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (or (<= U_m 1.75e-71) (and (not (<= U_m 1.9e-29)) (<= U_m 0.72)))
    (* -2.0 J_m)
    (* -2.0 (* U_m 0.5)))))
U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if ((U_m <= 1.75e-71) || (!(U_m <= 1.9e-29) && (U_m <= 0.72))) {
		tmp = -2.0 * J_m;
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return J_s * tmp;
}

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if ((u_m <= 1.75d-71) .or. (.not. (u_m <= 1.9d-29)) .and. (u_m <= 0.72d0)) then
        tmp = (-2.0d0) * j_m
    else
        tmp = (-2.0d0) * (u_m * 0.5d0)
    end if
    code = j_s * tmp
end function

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if ((U_m <= 1.75e-71) || (!(U_m <= 1.9e-29) && (U_m <= 0.72))) {
		tmp = -2.0 * J_m;
	} else {
		tmp = -2.0 * (U_m * 0.5);
	}
	return J_s * tmp;
}

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if (U_m <= 1.75e-71) or (not (U_m <= 1.9e-29) and (U_m <= 0.72)):
		tmp = -2.0 * J_m
	else:
		tmp = -2.0 * (U_m * 0.5)
	return J_s * tmp

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if ((U_m <= 1.75e-71) || (!(U_m <= 1.9e-29) && (U_m <= 0.72)))
		tmp = Float64(-2.0 * J_m);
	else
		tmp = Float64(-2.0 * Float64(U_m * 0.5));
	end
	return Float64(J_s * tmp)
end

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if ((U_m <= 1.75e-71) || (~((U_m <= 1.9e-29)) && (U_m <= 0.72)))
		tmp = -2.0 * J_m;
	else
		tmp = -2.0 * (U_m * 0.5);
	end
	tmp_2 = J_s * tmp;
end

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[Or[LessEqual[U$95$m, 1.75e-71], And[N[Not[LessEqual[U$95$m, 1.9e-29]], $MachinePrecision], LessEqual[U$95$m, 0.72]]], N[(-2.0 * J$95$m), $MachinePrecision], N[(-2.0 * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

\begin{array}{l}
U_m = \left|U\right|
\\
J_m = \left|J\right|
\\
J_s = \mathsf{copysign}\left(1, J\right)

\\
J_s \cdot \begin{array}{l}
\mathbf{if}\;U_m \leq 1.75 \cdot 10^{-71} \lor \neg \left(U_m \leq 1.9 \cdot 10^{-29}\right) \land U_m \leq 0.72:\\
\;\;\;\;-2 \cdot J_m\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(U_m \cdot 0.5\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 29.2% accurate, 140.0× speedup?

\[\begin{array}{l} U_m = \left|U\right| \\ J_m = \left|J\right| \\ J_s = \mathsf{copysign}\left(1, J\right) \\ J_s \cdot \left(-2 \cdot J_m\right) \end{array} \]
U_m = (fabs.f64 U)
J_m = (fabs.f64 J)
J_s = (copysign.f64 1 J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (* -2.0 J_m)))
U_m = fabs(U);
J_m = fabs(J);
J_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (-2.0 * J_m);
}

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0d0, J)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * ((-2.0d0) * j_m)
end function

U_m = Math.abs(U);
J_m = Math.abs(J);
J_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (-2.0 * J_m);
}

U_m = math.fabs(U)
J_m = math.fabs(J)
J_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * (-2.0 * J_m)

U_m = abs(U)
J_m = abs(J)
J_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(-2.0 * J_m))
end

U_m = abs(U);
J_m = abs(J);
J_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * (-2.0 * J_m);
end

U_m = N[Abs[U], $MachinePrecision]
J_m = N[Abs[J], $MachinePrecision]
J_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
U_m = \left|U\right|
\\
J_m = \left|J\right|
\\
J_s = \mathsf{copysign}\left(1, J\right)

\\
J_s \cdot \left(-2 \cdot J_m\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023350 
(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)))))