Toniolo and Linder, Equation (7)

Average Error: 42.9 → 13.7

Time: 30.3s

Precision: binary64

Cost: 21516

Math

\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \frac{t}{\left|t\right|}\\ t_3 := \sqrt{2} \cdot t\\ t_4 := \sqrt{\frac{2}{x}}\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{t}{\left(\left(-\ell\right) \cdot t_4\right) \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\ell \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{t_3}{\sqrt{\left(t_1 + \left(\ell \cdot \ell - \left(-\ell \cdot \ell\right)\right) \cdot {\left(\frac{1}{x}\right)}^{2}\right) - \left(-t_1\right)}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+51}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+140}:\\ \;\;\;\;\frac{t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell + \left(t \cdot t\right) \cdot 2}{x} + t \cdot t\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+177}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{\ell \cdot t_4}\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x))
        (t_2 (/ t (fabs t)))
        (t_3 (* (sqrt 2.0) t))
        (t_4 (sqrt (/ 2.0 x))))
   (if (<= l -9.2e+164)
     (/ t (* (* (- l) t_4) (sqrt 0.5)))
     (if (<= l -4.5e+130)
       t_2
       (if (<= l -8.2e+108)
         (/
          t_3
          (sqrt
           (-
            (+ t_1 (* (- (* l l) (- (* l l))) (pow (/ 1.0 x) 2.0)))
            (- t_1))))
         (if (<= l 1.45e+51)
           t_2
           (if (<= l 1.15e+140)
             (/
              t
              (*
               (sqrt (* 2.0 (+ (/ (+ (* l l) (* (* t t) 2.0)) x) (* t t))))
               (sqrt 0.5)))
             (if (<= l 3.9e+177) t_2 (/ t_3 (* l t_4))))))))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double t_2 = t / fabs(t);
	double t_3 = sqrt(2.0) * t;
	double t_4 = sqrt((2.0 / x));
	double tmp;
	if (l <= -9.2e+164) {
		tmp = t / ((-l * t_4) * sqrt(0.5));
	} else if (l <= -4.5e+130) {
		tmp = t_2;
	} else if (l <= -8.2e+108) {
		tmp = t_3 / sqrt(((t_1 + (((l * l) - -(l * l)) * pow((1.0 / x), 2.0))) - -t_1));
	} else if (l <= 1.45e+51) {
		tmp = t_2;
	} else if (l <= 1.15e+140) {
		tmp = t / (sqrt((2.0 * ((((l * l) + ((t * t) * 2.0)) / x) + (t * t)))) * sqrt(0.5));
	} else if (l <= 3.9e+177) {
		tmp = t_2;
	} else {
		tmp = t_3 / (l * t_4);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

↓

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (l * l) / x
    t_2 = t / abs(t)
    t_3 = sqrt(2.0d0) * t
    t_4 = sqrt((2.0d0 / x))
    if (l <= (-9.2d+164)) then
        tmp = t / ((-l * t_4) * sqrt(0.5d0))
    else if (l <= (-4.5d+130)) then
        tmp = t_2
    else if (l <= (-8.2d+108)) then
        tmp = t_3 / sqrt(((t_1 + (((l * l) - -(l * l)) * ((1.0d0 / x) ** 2.0d0))) - -t_1))
    else if (l <= 1.45d+51) then
        tmp = t_2
    else if (l <= 1.15d+140) then
        tmp = t / (sqrt((2.0d0 * ((((l * l) + ((t * t) * 2.0d0)) / x) + (t * t)))) * sqrt(0.5d0))
    else if (l <= 3.9d+177) then
        tmp = t_2
    else
        tmp = t_3 / (l * t_4)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

public static double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double t_2 = t / Math.abs(t);
	double t_3 = Math.sqrt(2.0) * t;
	double t_4 = Math.sqrt((2.0 / x));
	double tmp;
	if (l <= -9.2e+164) {
		tmp = t / ((-l * t_4) * Math.sqrt(0.5));
	} else if (l <= -4.5e+130) {
		tmp = t_2;
	} else if (l <= -8.2e+108) {
		tmp = t_3 / Math.sqrt(((t_1 + (((l * l) - -(l * l)) * Math.pow((1.0 / x), 2.0))) - -t_1));
	} else if (l <= 1.45e+51) {
		tmp = t_2;
	} else if (l <= 1.15e+140) {
		tmp = t / (Math.sqrt((2.0 * ((((l * l) + ((t * t) * 2.0)) / x) + (t * t)))) * Math.sqrt(0.5));
	} else if (l <= 3.9e+177) {
		tmp = t_2;
	} else {
		tmp = t_3 / (l * t_4);
	}
	return tmp;
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

↓

def code(x, l, t):
	t_1 = (l * l) / x
	t_2 = t / math.fabs(t)
	t_3 = math.sqrt(2.0) * t
	t_4 = math.sqrt((2.0 / x))
	tmp = 0
	if l <= -9.2e+164:
		tmp = t / ((-l * t_4) * math.sqrt(0.5))
	elif l <= -4.5e+130:
		tmp = t_2
	elif l <= -8.2e+108:
		tmp = t_3 / math.sqrt(((t_1 + (((l * l) - -(l * l)) * math.pow((1.0 / x), 2.0))) - -t_1))
	elif l <= 1.45e+51:
		tmp = t_2
	elif l <= 1.15e+140:
		tmp = t / (math.sqrt((2.0 * ((((l * l) + ((t * t) * 2.0)) / x) + (t * t)))) * math.sqrt(0.5))
	elif l <= 3.9e+177:
		tmp = t_2
	else:
		tmp = t_3 / (l * t_4)
	return tmp

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	t_2 = Float64(t / abs(t))
	t_3 = Float64(sqrt(2.0) * t)
	t_4 = sqrt(Float64(2.0 / x))
	tmp = 0.0
	if (l <= -9.2e+164)
		tmp = Float64(t / Float64(Float64(Float64(-l) * t_4) * sqrt(0.5)));
	elseif (l <= -4.5e+130)
		tmp = t_2;
	elseif (l <= -8.2e+108)
		tmp = Float64(t_3 / sqrt(Float64(Float64(t_1 + Float64(Float64(Float64(l * l) - Float64(-Float64(l * l))) * (Float64(1.0 / x) ^ 2.0))) - Float64(-t_1))));
	elseif (l <= 1.45e+51)
		tmp = t_2;
	elseif (l <= 1.15e+140)
		tmp = Float64(t / Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(l * l) + Float64(Float64(t * t) * 2.0)) / x) + Float64(t * t)))) * sqrt(0.5)));
	elseif (l <= 3.9e+177)
		tmp = t_2;
	else
		tmp = Float64(t_3 / Float64(l * t_4));
	end
	return tmp
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

↓

function tmp_2 = code(x, l, t)
	t_1 = (l * l) / x;
	t_2 = t / abs(t);
	t_3 = sqrt(2.0) * t;
	t_4 = sqrt((2.0 / x));
	tmp = 0.0;
	if (l <= -9.2e+164)
		tmp = t / ((-l * t_4) * sqrt(0.5));
	elseif (l <= -4.5e+130)
		tmp = t_2;
	elseif (l <= -8.2e+108)
		tmp = t_3 / sqrt(((t_1 + (((l * l) - -(l * l)) * ((1.0 / x) ^ 2.0))) - -t_1));
	elseif (l <= 1.45e+51)
		tmp = t_2;
	elseif (l <= 1.15e+140)
		tmp = t / (sqrt((2.0 * ((((l * l) + ((t * t) * 2.0)) / x) + (t * t)))) * sqrt(0.5));
	elseif (l <= 3.9e+177)
		tmp = t_2;
	else
		tmp = t_3 / (l * t_4);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -9.2e+164], N[(t / N[(N[((-l) * t$95$4), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.5e+130], t$95$2, If[LessEqual[l, -8.2e+108], N[(t$95$3 / N[Sqrt[N[(N[(t$95$1 + N[(N[(N[(l * l), $MachinePrecision] - (-N[(l * l), $MachinePrecision])), $MachinePrecision] * N[Power[N[(1.0 / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - (-t$95$1)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+51], t$95$2, If[LessEqual[l, 1.15e+140], N[(t / N[(N[Sqrt[N[(2.0 * N[(N[(N[(N[(l * l), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.9e+177], t$95$2, N[(t$95$3 / N[(l * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -9.1999999999999998e164

   1. Initial program 64.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Taylor expanded in x around inf 50.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]

   3. Simplified 50.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right)\right) - \left(-\frac{\mathsf{fma}\left(\ell, \ell, \left(t + t\right) \cdot t\right)}{x}\right)}}} \]
      Proof

   4. Taylor expanded in l around -inf 23.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-1 \cdot \left(\ell \cdot \sqrt{\frac{2}{x}}\right)}} \]

   5. Simplified 23.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(-\ell\right) \cdot \sqrt{\frac{2}{x}}}} \]
      Proof

   6. Applied egg-rr 23.7

      \[\leadsto \color{blue}{1 \cdot \frac{t}{\left(\left(-\ell\right) \cdot \sqrt{\frac{2}{x}}\right) \cdot \sqrt{0.5}}} \]

   7. Simplified 23.7

      \[\leadsto \color{blue}{\frac{t}{\left(\left(-\ell\right) \cdot \sqrt{\frac{2}{x}}\right) \cdot \sqrt{0.5}}} \]
      Proof

   if -9.1999999999999998e164 < l < -4.50000000000000039e130 or -8.1999999999999998e108 < l < 1.4499999999999999e51 or 1.14999999999999995e140 < l < 3.8999999999999999e177

   1. Initial program 36.9

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 36.9

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\mathsf{fma}\left(\frac{-1 - x}{1 - x}, \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right), -\ell \cdot \ell\right)} \cdot \sqrt{0.5}}} \]
      Proof

   3. Taylor expanded in x around inf 33.5

      \[\leadsto \frac{t}{\color{blue}{\sqrt{2 \cdot {t}^{2}}} \cdot \sqrt{0.5}} \]

   4. Simplified 33.5

      \[\leadsto \frac{t}{\color{blue}{\sqrt{\left(t \cdot t\right) \cdot 2}} \cdot \sqrt{0.5}} \]
      Proof

   5. Applied egg-rr 10.0

      \[\leadsto \frac{t}{\color{blue}{1 \cdot \frac{\left|t\right| \cdot \sqrt{2}}{\sqrt{2}}}} \]

   6. Simplified 10.0

      \[\leadsto \frac{t}{\color{blue}{\frac{\left|t\right| \cdot \sqrt{2}}{\sqrt{2}}}} \]
      Proof

   7. Taylor expanded in t around 0 9.8

      \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   if -4.50000000000000039e130 < l < -8.1999999999999998e108

   1. Initial program 56.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Taylor expanded in t around 0 59.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{\left(1 + x\right) \cdot {\ell}^{2}}{x - 1} - {\ell}^{2}}}} \]

   3. Simplified 59.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot \left(1 + x\right)\right)}{x - 1} - \ell \cdot \ell}}} \]
      Proof

   4. Taylor expanded in x around inf 39.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left({\ell}^{2} - -1 \cdot {\ell}^{2}\right) \cdot {\left(\frac{1}{x}\right)}^{2}\right) - -1 \cdot \frac{{\ell}^{2}}{x}}}} \]

   5. Simplified 39.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{\ell \cdot \ell}{x} + \left(\ell \cdot \ell - \left(-\ell \cdot \ell\right)\right) \cdot {\left(\frac{1}{x}\right)}^{2}\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}}} \]
      Proof

   if 1.4499999999999999e51 < l < 1.14999999999999995e140

   1. Initial program 52.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 52.0

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\mathsf{fma}\left(\frac{-1 - x}{1 - x}, \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right), -\ell \cdot \ell\right)} \cdot \sqrt{0.5}}} \]
      Proof

   3. Taylor expanded in x around inf 21.7

      \[\leadsto \frac{t}{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \cdot \sqrt{0.5}} \]

   4. Simplified 21.7

      \[\leadsto \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell \cdot \ell + \left(t \cdot t\right) \cdot 2}{x} + t \cdot t\right)}} \cdot \sqrt{0.5}} \]
      Proof

   if 3.8999999999999999e177 < l

   1. Initial program 64.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Taylor expanded in x around inf 48.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]

   3. Simplified 48.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right)\right) - \left(-\frac{\mathsf{fma}\left(\ell, \ell, \left(t + t\right) \cdot t\right)}{x}\right)}}} \]
      Proof

   4. Taylor expanded in l around inf 24.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x}}}} \]
3. Recombined 5 regimes into one program.

Alternatives

Alternative 1
Error	13.6
Cost	14804

\[\begin{array}{l} t_1 := \frac{t}{\left|t\right|}\\ t_2 := \sqrt{\frac{2}{x}}\\ t_3 := \left(-\ell\right) \cdot t_2\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{+164}:\\ \;\;\;\;\frac{t}{t_3 \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sqrt{2}}{t_3} \cdot t\\ \mathbf{elif}\;\ell \leq 1.46 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;\frac{t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell + \left(t \cdot t\right) \cdot 2}{x} + t \cdot t\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot t_2}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	13904

\[\begin{array}{l} t_1 := \frac{t}{\left|t\right|}\\ t_2 := \sqrt{\frac{2}{x}}\\ t_3 := \frac{\sqrt{2}}{\left(-\ell\right) \cdot t_2} \cdot t\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+166}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -5.4 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.95 \cdot 10^{+108}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot t_2}\\ \end{array} \]

Alternative 3
Error	13.6
Cost	13904

\[\begin{array}{l} t_1 := \frac{t}{\left|t\right|}\\ t_2 := \sqrt{\frac{2}{x}}\\ t_3 := \left(-\ell\right) \cdot t_2\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+167}:\\ \;\;\;\;\frac{t}{t_3 \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\ell \leq -4.2 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{2}}{t_3} \cdot t\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+174}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot t_2}\\ \end{array} \]

Alternative 4
Error	16.6
Cost	13640

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-185}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{t}\\ \end{array} \]

Alternative 5
Error	15.9
Cost	6592

\[\frac{t}{\left|t\right|} \]

Alternative 6
Error	39.1
Cost	192

\[\frac{t}{t} \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))