Toniolo and Linder, Equation (7)

Average Accuracy: 32.8% → 82.9%

Time: 29.8s

Precision: binary64

Cost: 14544

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-39}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-289}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

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
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt (* 2.0 (+ (* l (/ l x)) (* t (+ t (/ t x))))))))))
   (if (<= t -1.45e-39)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (<= t -1.7e-183)
       t_1
       (if (<= t -1.15e-289)
         -1.0
         (if (<= t 2.7e+98) t_1 (+ 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 = t * (sqrt(2.0) / sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	double tmp;
	if (t <= -1.45e-39) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -1.7e-183) {
		tmp = t_1;
	} else if (t <= -1.15e-289) {
		tmp = -1.0;
	} else if (t <= 2.7e+98) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (sqrt(2.0d0) / sqrt((2.0d0 * ((l * (l / x)) + (t * (t + (t / x)))))))
    if (t <= (-1.45d-39)) then
        tmp = -sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else if (t <= (-1.7d-183)) then
        tmp = t_1
    else if (t <= (-1.15d-289)) then
        tmp = -1.0d0
    else if (t <= 2.7d+98) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double l, double t) {
	double t_1 = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	double tmp;
	if (t <= -1.45e-39) {
		tmp = -Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else if (t <= -1.7e-183) {
		tmp = t_1;
	} else if (t <= -1.15e-289) {
		tmp = -1.0;
	} else if (t <= 2.7e+98) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

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

↓

def code(x, l, t):
	t_1 = t * (math.sqrt(2.0) / math.sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))))
	tmp = 0
	if t <= -1.45e-39:
		tmp = -math.sqrt(((-1.0 + x) / (x + 1.0)))
	elif t <= -1.7e-183:
		tmp = t_1
	elif t <= -1.15e-289:
		tmp = -1.0
	elif t <= 2.7e+98:
		tmp = t_1
	else:
		tmp = 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(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(l * Float64(l / x)) + Float64(t * Float64(t + Float64(t / x))))))))
	tmp = 0.0
	if (t <= -1.45e-39)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif (t <= -1.7e-183)
		tmp = t_1;
	elseif (t <= -1.15e-289)
		tmp = -1.0;
	elseif (t <= 2.7e+98)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(2.0) / sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	tmp = 0.0;
	if (t <= -1.45e-39)
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	elseif (t <= -1.7e-183)
		tmp = t_1;
	elseif (t <= -1.15e-289)
		tmp = -1.0;
	elseif (t <= 2.7e+98)
		tmp = t_1;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.45e-39], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1.7e-183], t$95$1, If[LessEqual[t, -1.15e-289], -1.0, If[LessEqual[t, 2.7e+98], t$95$1, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.44999999999999994e-39

   1. Initial program 38.2%

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

   2. Simplified 38.2%

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

      [Start] 38.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 38.2	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around -inf 91.0%

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

   4. Simplified 91.0%

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

      [Start] 91.0	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      associate-*r* [=>] 91.0	\[ \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      *-commutative [<=] 91.0	\[ \frac{\sqrt{2}}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      neg-mul-1 [<=] 91.0	\[ \frac{\sqrt{2}}{\color{blue}{\left(-t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      distribute-rgt-neg-in [=>] 91.0	\[ \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      +-commutative [<=] 91.0	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \cdot t \]
      sub-neg [=>] 91.0	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 91.0	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 91.0	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Taylor expanded in t around 0 91.4%

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

   6. Simplified 91.4%

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

      [Start] 91.4	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 91.4	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 91.4	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 91.4	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [=>] 91.4	\[ -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]
      +-commutative [=>] 91.4	\[ -\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \]

   if -1.44999999999999994e-39 < t < -1.70000000000000007e-183 or -1.1500000000000001e-289 < t < 2.7e98

   1. Initial program 39.6%

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

   2. Simplified 39.7%

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

      [Start] 39.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 39.7	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 72.8%

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

   4. Simplified 72.8%

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

      [Start] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 72.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 72.4%

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

   6. Simplified 72.4%

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

      [Start] 72.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 72.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 78.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25}}} \cdot t \]
      Proof

      [Start] 72.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      pow1/2 [=>] 72.5	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{0.5}}} \cdot t \]
      sqr-pow [=>] 72.3	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot t \]

   8. Simplified 78.9%

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

      [Start] 78.7	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 78.9	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 78.9	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 78.9	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)}}} \cdot t \]
      fma-udef [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right) + \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right) + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}}} \cdot t \]
      fma-udef [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      count-2 [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}}} \cdot t \]
      *-commutative [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}} \cdot t \]
      fma-udef [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(\frac{t}{x} \cdot t + t \cdot t\right)}\right)}} \cdot t \]
      +-commutative [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right)}} \cdot t \]
      distribute-rgt-out [=>] 78.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{t \cdot \left(t + \frac{t}{x}\right)}\right)}} \cdot t \]

   if -1.70000000000000007e-183 < t < -1.1500000000000001e-289

   1. Initial program 1.3%

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

   2. Simplified 1.3%

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

      [Start] 1.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 1.3	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around -inf 37.5%

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

   4. Simplified 37.5%

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

      [Start] 37.5	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      associate-*r* [=>] 37.5	\[ \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      *-commutative [<=] 37.5	\[ \frac{\sqrt{2}}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      neg-mul-1 [<=] 37.5	\[ \frac{\sqrt{2}}{\color{blue}{\left(-t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      distribute-rgt-neg-in [=>] 37.5	\[ \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      +-commutative [<=] 37.5	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \cdot t \]
      sub-neg [=>] 37.5	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 37.5	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 37.5	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Taylor expanded in x around inf 37.4%

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

   if 2.7e98 < t

   1. Initial program 21.1%

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

   2. Simplified 21.2%

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

      [Start] 21.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 21.2	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around -inf 1.6%

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

   4. Simplified 1.6%

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

      [Start] 1.6	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      associate-*r* [=>] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      *-commutative [<=] 1.6	\[ \frac{\sqrt{2}}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      neg-mul-1 [<=] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{\left(-t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      distribute-rgt-neg-in [=>] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      +-commutative [<=] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \cdot t \]
      sub-neg [=>] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 1.6	\[ \frac{\sqrt{2}}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Applied egg-rr 22.4%

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

      No proof available- proof too large to flatten.

   6. Simplified 96.2%

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

      [Start] 22.4	\[ \sqrt{\left(t \cdot t\right) \cdot \frac{2}{\frac{x + 1}{x + -1} \cdot \left(2 \cdot \left(t \cdot t\right)\right)}} \]
      associate-*r/ [=>] 22.5	\[ \sqrt{\color{blue}{\frac{\left(t \cdot t\right) \cdot 2}{\frac{x + 1}{x + -1} \cdot \left(2 \cdot \left(t \cdot t\right)\right)}}} \]
      *-commutative [<=] 22.5	\[ \sqrt{\frac{\color{blue}{2 \cdot \left(t \cdot t\right)}}{\frac{x + 1}{x + -1} \cdot \left(2 \cdot \left(t \cdot t\right)\right)}} \]
      *-commutative [=>] 22.5	\[ \sqrt{\frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1}}}} \]
      associate-/r* [=>] 22.5	\[ \sqrt{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot t\right)}{2 \cdot \left(t \cdot t\right)}}{\frac{x + 1}{x + -1}}}} \]
      *-inverses [=>] 96.2	\[ \sqrt{\frac{\color{blue}{1}}{\frac{x + 1}{x + -1}}} \]
      +-commutative [=>] 96.2	\[ \sqrt{\frac{1}{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 95.1%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-39}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-183}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-289}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+98}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	76.2%
Cost	7504

\[\begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-91}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-290}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-184}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 2
Accuracy	76.0%
Cost	7376

\[\begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-290}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 3
Accuracy	76.2%
Cost	7376

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{-91}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-289}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-186}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{-\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	75.7%
Cost	7312

\[\begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-290}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-185}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 5
Accuracy	75.7%
Cost	7312

\[\begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{elif}\;t \leq -1.95 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-291}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{-\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 6
Accuracy	75.8%
Cost	7248

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{x}}{\ell}\\ t_2 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_2\right)\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-290}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_2 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 7
Accuracy	75.8%
Cost	7248

\[\begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-289}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-192}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 8
Accuracy	75.8%
Cost	836

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

Alternative 9
Accuracy	75.9%
Cost	836

\[\begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - t_1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Accuracy	75.3%
Cost	452

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

Alternative 11
Accuracy	75.6%
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 12
Accuracy	74.9%
Cost	196

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

Alternative 13
Accuracy	38.7%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023152 
(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)))))