Toniolo and Linder, Equation (7)

Average Error: 43.1 → 12.5

Time: 35.4s

Precision: binary64

Cost: 48712

Math

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

↓

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\\ t_2 := \ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\\ t_3 := \sqrt{2} \cdot t\\ t_4 := \frac{t_3}{\sqrt{\frac{x + 1}{x + -1} \cdot t_2 - \ell \cdot \ell}}\\ \mathbf{if}\;t_4 \leq -5 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(-1, \frac{t_1 + t_1}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right) + \frac{t_2}{x}\right)}}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_3 \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \end{array} \]

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 (- (* (* t t) -2.0) (* l l)))
        (t_2 (+ (* l l) (* 2.0 (* t t))))
        (t_3 (* (sqrt 2.0) t))
        (t_4 (/ t_3 (sqrt (- (* (/ (+ x 1.0) (+ x -1.0)) t_2) (* l l))))))
   (if (<= t_4 -5e-257)
     (*
      (sqrt 2.0)
      (/
       t
       (sqrt
        (+
         (/ (* l l) x)
         (+
          (fma -1.0 (/ (+ t_1 t_1) (* x x)) (* 2.0 (+ (* t t) (/ (* t t) x))))
          (/ t_2 x))))))
     (if (<= t_4 INFINITY)
       (* t_3 (/ 1.0 (hypot t t)))
       (/ t_3 (fabs (* l (sqrt (/ 2.0 x)))))))))

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 = ((t * t) * -2.0) - (l * l);
	double t_2 = (l * l) + (2.0 * (t * t));
	double t_3 = sqrt(2.0) * t;
	double t_4 = t_3 / sqrt(((((x + 1.0) / (x + -1.0)) * t_2) - (l * l)));
	double tmp;
	if (t_4 <= -5e-257) {
		tmp = sqrt(2.0) * (t / sqrt((((l * l) / x) + (fma(-1.0, ((t_1 + t_1) / (x * x)), (2.0 * ((t * t) + ((t * t) / x)))) + (t_2 / x)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3 * (1.0 / hypot(t, t));
	} else {
		tmp = t_3 / fabs((l * sqrt((2.0 / x))));
	}
	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(Float64(t * t) * -2.0) - Float64(l * l))
	t_2 = Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))
	t_3 = Float64(sqrt(2.0) * t)
	t_4 = Float64(t_3 / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x + -1.0)) * t_2) - Float64(l * l))))
	tmp = 0.0
	if (t_4 <= -5e-257)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(fma(-1.0, Float64(Float64(t_1 + t_1) / Float64(x * x)), Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))) + Float64(t_2 / x))))));
	elseif (t_4 <= Inf)
		tmp = Float64(t_3 * Float64(1.0 / hypot(t, t)));
	else
		tmp = Float64(t_3 / abs(Float64(l * sqrt(Float64(2.0 / x)))));
	end
	return 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[(N[(t * t), $MachinePrecision] * -2.0), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-257], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-1.0 * N[(N[(t$95$1 + t$95$1), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$3 * N[(1.0 / N[Sqrt[t ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[Abs[N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < -4.99999999999999989e-257

   1. Initial program 28.7

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

   2. Simplified 34.7

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 4 points increase in error, 24 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 1 points increase in error, 15 points decrease in error

   3. Taylor expanded in x around -inf 11.1

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

   4. Simplified 11.1

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(-1, \frac{\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right) - \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x \cdot x}, 2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right)\right) - \frac{-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (*.f64 t t)))) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (neg.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (*.f64 t t)))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) (*.f64 x x)) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2)) (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2)) (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (fma.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2)) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (neg.f64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (*.f64 t t)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   if -4.99999999999999989e-257 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 43.3

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

   2. Taylor expanded in x around 0 63.1

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

   3. Simplified 63.1

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\left(-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)\right) - \ell \cdot \ell}} \cdot \left(t \cdot \sqrt{2}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 l l)))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (neg.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))) (*.f64 l l)))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) (*.f64 l l)))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 l l)))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 l l)))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= unpow2_binary64 (pow.f64 l 2))))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 l 2)))) (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 1 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 l 2)))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 44.1

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

   5. Simplified 8.3

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(t, t\right)}} \cdot \left(t \cdot \sqrt{2}\right) \]
      Proof
      (/.f64 1 (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 t t) (*.f64 t t))))): 68 points increase in error, 43 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite=> count-2_binary64 (*.f64 2 (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 2 (*.f64 t t)) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (+.f64 (*.f64 2 (*.f64 t t)) (Rewrite<= +-inverses_binary64 (-.f64 (pow.f64 l 2) (pow.f64 l 2)))))): 36 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 2 (*.f64 t t)) (pow.f64 l 2)) (pow.f64 l 2))))): 37 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (-.f64 (+.f64 (*.f64 2 (*.f64 t t)) (Rewrite=> unpow2_binary64 (*.f64 l l))) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (-.f64 (Rewrite<= fma-udef_binary64 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (sqrt.f64 (-.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) (Rewrite=> unpow2_binary64 (*.f64 l l))))): 0 points increase in error, 0 points decrease in error

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l 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 49.3

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

   3. Simplified 49.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in l around inf 37.5

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

   5. Simplified 37.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}} \]
      Proof
      (*.f64 l (*.f64 (sqrt.f64 2) (sqrt.f64 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l (sqrt.f64 2)) (sqrt.f64 (/.f64 1 x)))): 30 points increase in error, 29 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) l)) (sqrt.f64 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 48.9

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

   7. Simplified 26.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}} \]
      Proof
      (fabs.f64 (*.f64 l (sqrt.f64 (/.f64 2 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 (*.f64 l (sqrt.f64 (/.f64 2 x))) (*.f64 l (sqrt.f64 (/.f64 2 x)))))): 41 points increase in error, 12 points decrease in error
      (sqrt.f64 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 l l) (*.f64 (sqrt.f64 (/.f64 2 x)) (sqrt.f64 (/.f64 2 x)))))): 26 points increase in error, 9 points decrease in error
      (sqrt.f64 (*.f64 (*.f64 l l) (Rewrite=> rem-square-sqrt_binary64 (/.f64 2 x)))): 13 points increase in error, 64 points decrease in error
      (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 (*.f64 l l) (/.f64 2 x)) 1/2)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq -5 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(-1, \frac{\left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right) + \left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right)}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	48712

\[\begin{array}{l} t_1 := \ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\\ t_2 := \sqrt{2} \cdot t\\ t_3 := \frac{t_2}{\sqrt{\frac{x + 1}{x + -1} \cdot t_1 - \ell \cdot \ell}}\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-257}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{t_1}{x}\right)}}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_2 \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \end{array} \]

Alternative 2
Error	12.5
Cost	48712

\[\begin{array}{l} t_1 := \sqrt{2} \cdot t\\ t_2 := \frac{t_1}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_1 \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \end{array} \]

Alternative 3
Error	13.8
Cost	14924

\[\begin{array}{l} t_1 := \sqrt{2} \cdot t\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+190}:\\ \;\;\;\;\frac{t}{\ell} \cdot \left(-\sqrt{x}\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;t_1 \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+146}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \end{array} \]

Alternative 4
Error	14.0
Cost	13576

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{t}{\ell} \cdot \left(-\sqrt{x}\right)\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+145}:\\ \;\;\;\;\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \end{array} \]

Alternative 5
Error	13.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-164}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-294}:\\ \;\;\;\;\frac{t}{\sqrt{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 6
Error	13.6
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-165}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-283}:\\ \;\;\;\;\frac{t}{\sqrt{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7
Error	13.5
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{-164}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-282}:\\ \;\;\;\;\frac{t}{\sqrt{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	14.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-260}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-282}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 9
Error	14.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-257}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 4.15 \cdot 10^{-283}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 10
Error	15.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-295}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 11
Error	14.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-295}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 12
Error	15.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-295}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13
Error	15.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-295}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 14
Error	15.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-295}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	38.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022318 
(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)))))