Toniolo and Linder, Equation (7)

Average Error: 43.4 → 12.8

Time: 33.3s

Precision: binary64

Cost: 23376

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_3 := t \cdot \sqrt{2}\\ t_4 := \left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+61}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_2 + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-233}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-84}:\\ \;\;\;\;\frac{t_3}{\sqrt{\left(t_1 + \mathsf{fma}\left(-1, \frac{t_4 + t_4}{x \cdot x}, t_2\right)\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t_3 \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

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 (* 2.0 (+ (* t t) (/ (* t t) x))))
        (t_3 (* t (sqrt 2.0)))
        (t_4 (- (* (* t t) -2.0) (* l l))))
   (if (<= t -2.5e+61)
     -1.0
     (if (<= t -1.45e-183)
       (*
        t
        (/ (sqrt 2.0) (sqrt (+ t_1 (+ t_2 (/ (fma 2.0 (* t t) (* l l)) x))))))
       (if (<= t 1.32e-233)
         (/ (/ t (pow x -0.5)) l)
         (if (<= t 1.65e-84)
           (/
            t_3
            (sqrt
             (+
              (+ t_1 (fma -1.0 (/ (+ t_4 t_4) (* x x)) t_2))
              (/ (+ (* l l) (* 2.0 (* t t))) x))))
           (/ t_3 (* t_3 (sqrt (/ (+ x 1.0) (+ -1.0 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 = (l * l) / x;
	double t_2 = 2.0 * ((t * t) + ((t * t) / x));
	double t_3 = t * sqrt(2.0);
	double t_4 = ((t * t) * -2.0) - (l * l);
	double tmp;
	if (t <= -2.5e+61) {
		tmp = -1.0;
	} else if (t <= -1.45e-183) {
		tmp = t * (sqrt(2.0) / sqrt((t_1 + (t_2 + (fma(2.0, (t * t), (l * l)) / x)))));
	} else if (t <= 1.32e-233) {
		tmp = (t / pow(x, -0.5)) / l;
	} else if (t <= 1.65e-84) {
		tmp = t_3 / sqrt(((t_1 + fma(-1.0, ((t_4 + t_4) / (x * x)), t_2)) + (((l * l) + (2.0 * (t * t))) / x)));
	} else {
		tmp = t_3 / (t_3 * sqrt(((x + 1.0) / (-1.0 + 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(l * l) / x)
	t_2 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	t_3 = Float64(t * sqrt(2.0))
	t_4 = Float64(Float64(Float64(t * t) * -2.0) - Float64(l * l))
	tmp = 0.0
	if (t <= -2.5e+61)
		tmp = -1.0;
	elseif (t <= -1.45e-183)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_1 + Float64(t_2 + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))));
	elseif (t <= 1.32e-233)
		tmp = Float64(Float64(t / (x ^ -0.5)) / l);
	elseif (t <= 1.65e-84)
		tmp = Float64(t_3 / sqrt(Float64(Float64(t_1 + fma(-1.0, Float64(Float64(t_4 + t_4) / Float64(x * x)), t_2)) + Float64(Float64(Float64(l * l) + Float64(2.0 * Float64(t * t))) / x))));
	else
		tmp = Float64(t_3 / Float64(t_3 * sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + 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[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t * t), $MachinePrecision] * -2.0), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.5e+61], -1.0, If[LessEqual[t, -1.45e-183], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(t$95$2 + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.32e-233], N[(N[(t / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t, 1.65e-84], N[(t$95$3 / N[Sqrt[N[(N[(t$95$1 + N[(-1.0 * N[(N[(t$95$4 + t$95$4), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$3 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.50000000000000009e61

   1. Initial program 46.2

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

   2. Simplified 57.7

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}} \cdot t} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (fma.f64 (*.f64 2 t) t (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 2 t) t) (*.f64 l l)))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l)))) t): 24 points increase in error, 36 points decrease in error
      (Rewrite=> associate-*l/_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))))): 18 points increase in error, 8 points decrease in error

   3. Taylor expanded in x around inf 63.0

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

   4. Simplified 63.0

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2}}} \cdot t \]
      Proof
      (*.f64 t (sqrt.f64 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 42.8

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

   6. Simplified 42.8

      \[\leadsto \color{blue}{\sqrt{\frac{1}{t \cdot t}}} \cdot t \]
      Proof
      (sqrt.f64 (/.f64 1 (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 2 2)) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 2 (*.f64 t t))))): 0 points increase in error, 0 points decrease in error

   7. Taylor expanded in t around -inf 3.9

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

   if -2.50000000000000009e61 < t < -1.45e-183

   1. Initial program 32.5

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

   2. Simplified 34.3

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}} \cdot t} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (fma.f64 (*.f64 2 t) t (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 2 t) t) (*.f64 l l)))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l)))) t): 24 points increase in error, 36 points decrease in error
      (Rewrite=> associate-*l/_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))))): 18 points increase in error, 8 points decrease in error

   3. Taylor expanded in x around inf 12.9

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

   4. Simplified 12.9

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]
      Proof
      (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (fma.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 (fma.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 (fma.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 (fma.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 (fma.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 (Rewrite=> fma-udef_binary64 (+.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 (+.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 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 (Rewrite<= +-commutative_binary64 (+.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))) (/.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

   if -1.45e-183 < t < 1.32000000000000006e-233

   1. Initial program 63.1

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

   2. Taylor expanded in x around inf 30.7

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

   3. Simplified 30.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)}{x}\right)}}} \]
      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 34.3

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

   5. Simplified 34.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-\ell \cdot \sqrt{2}\right)}} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 1 x)) (neg.f64 (*.f64 l (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 1 x)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) l)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (/.f64 1 x)) (*.f64 (sqrt.f64 2) l)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) l) (sqrt.f64 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) l) (sqrt.f64 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 37.3

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

   7. Simplified 34.7

      \[\leadsto \color{blue}{\frac{\frac{t}{{x}^{-0.5}}}{\ell}} \]
      Proof
      (/.f64 (/.f64 t (pow.f64 x -1/2)) l): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (/.f64 t (pow.f64 x -1/2)) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 (sqrt.f64 2) (sqrt.f64 2))) (/.f64 (/.f64 t (pow.f64 x -1/2)) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sqrt.f64 2) (/.f64 t (pow.f64 x -1/2))) (*.f64 (sqrt.f64 2) l))): 20 points increase in error, 18 points decrease in error
      (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (pow.f64 x -1/2))) (*.f64 (sqrt.f64 2) l)): 23 points increase in error, 14 points decrease in error
      (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (sqrt.f64 2) (pow.f64 x -1/2)) t)) (*.f64 (sqrt.f64 2) l)): 17 points increase in error, 24 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (sqrt.f64 2) (pow.f64 x -1/2)) (/.f64 t (*.f64 (sqrt.f64 2) l)))): 21 points increase in error, 24 points decrease in error

   if 1.32000000000000006e-233 < t < 1.64999999999999992e-84

   1. Initial program 49.7

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

   2. Taylor expanded in x around -inf 23.6

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

   3. Simplified 23.6

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

   if 1.64999999999999992e-84 < t

   1. Initial program 38.4

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

   2. Taylor expanded in l around 0 44.5

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

   3. Simplified 35.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{x + 1}{\frac{-1 + x}{t \cdot t}}}}} \]
      Proof
      (*.f64 2 (/.f64 (+.f64 x 1) (/.f64 (+.f64 -1 x) (*.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 x)) (/.f64 (+.f64 -1 x) (*.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (+.f64 1 x) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)) (*.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (+.f64 1 x) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (*.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (+.f64 1 x) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (*.f64 t t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (+.f64 1 x) (/.f64 (-.f64 x 1) (Rewrite<= unpow2_binary64 (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (-.f64 x 1)))): 60 points increase in error, 27 points decrease in error

   4. Taylor expanded in t around 0 7.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+61}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-233}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-84}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell \cdot \ell}{x} + \mathsf{fma}\left(-1, \frac{\left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right) + \left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right)}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.8
Cost	23376

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\\ t_3 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_4 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+61}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-185}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_3 + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{t_1 + \left(\mathsf{fma}\left(-1, \frac{t_2 + t_2}{x \cdot x}, t_3\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{t_4 \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 2
Error	12.6
Cost	21448

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_3 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+61}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-185}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_2 + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-246}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-84}:\\ \;\;\;\;\frac{t_3}{\sqrt{t_1 + \left(t_2 + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t_3 \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 3
Error	12.5
Cost	20688

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{+61}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-182}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 4
Error	12.6
Cost	15440

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+61}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \sqrt{2 + \left(\frac{2}{x} + \left(\frac{2}{x} + \frac{4}{x \cdot x}\right)\right)}}\\ \end{array} \]

Alternative 5
Error	15.1
Cost	14408

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-174}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \sqrt{2 + \left(\frac{2}{x} + \left(\frac{2}{x} + \frac{4}{x \cdot x}\right)\right)}}\\ \end{array} \]

Alternative 6
Error	15.2
Cost	14280

\[\begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \sqrt{\frac{4}{x \cdot x} + \left(2 + 2 \cdot \frac{2}{x}\right)}}\\ \end{array} \]

Alternative 7
Error	15.0
Cost	13444

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{t_1}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{t_1}}\\ \end{array} \]

Alternative 8
Error	15.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.24 \cdot 10^{-172}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 9
Error	15.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-174}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	15.4
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-219}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	16.1
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	39.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022296 
(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)))))