Toniolo and Linder, Equation (7)

Average Error: 42.5 → 8.2

Time: 26.3s

Precision: binary64

Cost: 42124

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_2 := \frac{t_1}{x}\\ \mathbf{if}\;t \leq -80000000000000:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-21}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-133}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell \cdot \ell}{x} + \mathsf{fma}\left(-1, \frac{-\left(t_1 + t_1\right)}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)\right) + t_2}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-86}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{t_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

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 (fma 2.0 (* t t) (* l l))) (t_2 (/ t_1 x)))
   (if (<= t -80000000000000.0)
     (- (sqrt (/ (+ x -1.0) (+ x 1.0))))
     (if (<= t -5.9e-21)
       (/ t (hypot t (/ l (sqrt x))))
       (if (<= t -1.55e-133)
         (/
          (* t (sqrt 2.0))
          (sqrt
           (+
            (+
             (/ (* l l) x)
             (fma
              -1.0
              (/ (- (+ t_1 t_1)) (* x x))
              (* 2.0 (+ (* t t) (/ (* t t) x)))))
            t_2)))
         (if (<= t 1.45e-86)
           (/ t (hypot t (sqrt t_2)))
           (+ 1.0 (+ (/ (/ 0.5 x) x) (/ -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 = fma(2.0, (t * t), (l * l));
	double t_2 = t_1 / x;
	double tmp;
	if (t <= -80000000000000.0) {
		tmp = -sqrt(((x + -1.0) / (x + 1.0)));
	} else if (t <= -5.9e-21) {
		tmp = t / hypot(t, (l / sqrt(x)));
	} else if (t <= -1.55e-133) {
		tmp = (t * sqrt(2.0)) / sqrt(((((l * l) / x) + fma(-1.0, (-(t_1 + t_1) / (x * x)), (2.0 * ((t * t) + ((t * t) / x))))) + t_2));
	} else if (t <= 1.45e-86) {
		tmp = t / hypot(t, sqrt(t_2));
	} else {
		tmp = 1.0 + (((0.5 / x) / x) + (-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 = fma(2.0, Float64(t * t), Float64(l * l))
	t_2 = Float64(t_1 / x)
	tmp = 0.0
	if (t <= -80000000000000.0)
		tmp = Float64(-sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	elseif (t <= -5.9e-21)
		tmp = Float64(t / hypot(t, Float64(l / sqrt(x))));
	elseif (t <= -1.55e-133)
		tmp = Float64(Float64(t * sqrt(2.0)) / sqrt(Float64(Float64(Float64(Float64(l * l) / x) + fma(-1.0, Float64(Float64(-Float64(t_1 + t_1)) / Float64(x * x)), Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))) + t_2)));
	elseif (t <= 1.45e-86)
		tmp = Float64(t / hypot(t, sqrt(t_2)));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(0.5 / x) / x) + 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[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / x), $MachinePrecision]}, If[LessEqual[t, -80000000000000.0], (-N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -5.9e-21], N[(t / N[Sqrt[t ^ 2 + N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.55e-133], N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + 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]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e-86], N[(t / N[Sqrt[t ^ 2 + N[Sqrt[t$95$2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8e13

   1. Initial program 42.1

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

   2. Simplified 42.0

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

      [Start] 42.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 42.2	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 53.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 42.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 42.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 42.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 42.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 42.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 42.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 42.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around inf 52.3

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

   4. Simplified 52.3

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

      [Start] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \]
      *-commutative [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \]
      unpow2 [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + x\right)}{x - 1}}} \]
      +-commutative [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \]
      sub-neg [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 52.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}}}} \]

   5. Taylor expanded in t around -inf 4.6

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

   6. Simplified 4.6

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]
      Proof

      [Start] 4.6	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 4.6	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 4.6	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 4.6	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   if -8e13 < t < -5.9000000000000003e-21

   1. Initial program 23.6

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

   2. Simplified 24.3

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

      [Start] 23.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 23.6	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 23.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 24.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 24.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 24.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 24.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 24.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 24.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 24.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 9.7

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

   4. Simplified 9.7

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

      [Start] 9.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 9.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 9.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 9.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 9.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 9.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 9.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 10.2

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

   6. Simplified 10.2

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

      [Start] 10.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + t \cdot t\right)}} \]
      unpow2 [=>] 10.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + t \cdot t\right)}} \]

   7. Applied egg-rr 43.3

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)} - 1} \]

   8. Simplified 26.2

      \[\leadsto \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]
      Proof

      [Start] 43.3	\[ e^{\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)} - 1 \]
      expm1-def [=>] 29.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\right)\right)} \]
      expm1-log1p [=>] 26.3	\[ \color{blue}{\frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]
      associate-/l* [=>] 26.2	\[ \frac{\color{blue}{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)} \]
      associate-/l/ [=>] 26.2	\[ \color{blue}{\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right) \cdot \frac{\sqrt{2}}{\sqrt{2}}}} \]
      *-inverses [=>] 26.2	\[ \frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right) \cdot \color{blue}{1}} \]
      *-rgt-identity [=>] 26.2	\[ \frac{t}{\color{blue}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}} \]

   if -5.9000000000000003e-21 < t < -1.55000000000000008e-133

   1. Initial program 30.2

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

   2. Taylor expanded in x around -inf 9.6

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

   3. Simplified 9.6

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

      [Start] 9.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]

   if -1.55000000000000008e-133 < t < 1.45e-86

   1. Initial program 55.8

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

   2. Simplified 60.5

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

      [Start] 55.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 55.8	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 54.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 61.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 61.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 61.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 61.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 61.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 61.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 60.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 27.3

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

   4. Simplified 27.3

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

      [Start] 27.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 27.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 27.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 27.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 27.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 27.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 27.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Applied egg-rr 9.8

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

   6. Simplified 9.8

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

      [Start] 9.8	\[ \frac{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)} \]
      *-inverses [=>] 9.8	\[ \frac{\frac{t}{\color{blue}{1}}}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)} \]

   if 1.45e-86 < t

   1. Initial program 38.0

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

   2. Simplified 39.1

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

      [Start] 38.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 38.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 47.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 39.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 39.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 39.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 39.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 39.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 39.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 39.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around inf 44.2

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

   4. Simplified 44.2

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

      [Start] 44.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \]
      *-commutative [=>] 44.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \]
      unpow2 [=>] 44.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + x\right)}{x - 1}}} \]
      +-commutative [=>] 44.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \]
      sub-neg [=>] 44.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 44.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 44.2	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}}}} \]

   5. Taylor expanded in t around 0 7.7

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]

   6. Taylor expanded in x around inf 8.0

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]

   7. Simplified 8.0

      \[\leadsto \color{blue}{1 + \left(\frac{\frac{0.5}{x}}{x} - \frac{1}{x}\right)} \]
      Proof

      [Start] 8.0	\[ \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x} \]
      sub-neg [=>] 8.0	\[ \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{1}{x}\right)} \]
      associate-+l+ [=>] 8.0	\[ \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\frac{1}{x}\right)\right)} \]
      sub-neg [<=] 8.0	\[ 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]
      associate-*r/ [=>] 8.0	\[ 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]
      metadata-eval [=>] 8.0	\[ 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]
      unpow2 [=>] 8.0	\[ 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]
      associate-/r* [=>] 8.0	\[ 1 + \left(\color{blue}{\frac{\frac{0.5}{x}}{x}} - \frac{1}{x}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -80000000000000:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-21}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-133}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell \cdot \ell}{x} + \mathsf{fma}\left(-1, \frac{-\left(\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right)}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-86}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.8
Cost	20496

\[\begin{array}{l} t_1 := -\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -170000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-14}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-86}:\\ \;\;\;\;\frac{t}{\mathsf{hypot}\left(t, \sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 2
Error	10.1
Cost	14924

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{if}\;t \leq -1100000000000:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-160}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 3
Error	10.0
Cost	14156

\[\begin{array}{l} t_1 := \frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{if}\;t \leq -170000000000:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 4
Error	12.0
Cost	13712

\[\begin{array}{l} t_1 := \frac{t}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{if}\;t \leq -15200000000000:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-220}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 5
Error	14.0
Cost	7240

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{-244}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell} \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.0
Cost	7240

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -9 \cdot 10^{-245}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-278}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	14.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -2.75 \cdot 10^{-248}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-279}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 8
Error	14.2
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{-257}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-278}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	14.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-255}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Error	14.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-247}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-278}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 11
Error	14.9
Cost	1220

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 12
Error	15.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 13
Error	14.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 14
Error	15.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 15
Error	15.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 16
Error	15.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	39.2
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023059 
(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)))))