Toniolo and Linder, Equation (7)

Average Accuracy: 33.5% → 76.9%

Time: 29.7s

Precision: binary64

Cost: 7044

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq 7.8 \cdot 10^{-308}:\\ \;\;\;\;-t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t 7.8e-308) (- t_1) t_1)))

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 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= 7.8e-308) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	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 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= 7.8d-308) then
        tmp = -t_1
    else
        tmp = t_1
    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 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= 7.8e-308) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	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 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= 7.8e-308:
		tmp = -t_1
	else:
		tmp = t_1
	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 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= 7.8e-308)
		tmp = Float64(-t_1);
	else
		tmp = t_1;
	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 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= 7.8e-308)
		tmp = -t_1;
	else
		tmp = t_1;
	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[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 7.8e-308], (-t$95$1), t$95$1]]

Derivation

1. Split input into 2 regimes

2. if t < 7.7999999999999999e-308

   1. Initial program 29.5%

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

   2. Simplified 29.4%

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

      [Start] 29.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 29.4	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around -inf 79.0%

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

   4. Simplified 79.0%

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

      [Start] 79.0	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      mul-1-neg [=>] 79.0	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      distribute-lft-neg-in [=>] 79.0	\[ \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      distribute-rgt-neg-out [<=] 79.0	\[ \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      +-commutative [=>] 79.0	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \cdot t \]
      sub-neg [=>] 79.0	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 79.0	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 79.0	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Taylor expanded in t around 0 79.3%

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

   6. Simplified 79.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{x + 1}}} \]
      Step-by-step derivation

      [Start] 79.3	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 79.3	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 79.3	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 79.3	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [=>] 79.3	\[ -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]
      +-commutative [=>] 79.3	\[ -\sqrt{\frac{-1 + x}{\color{blue}{x + 1}}} \]

   if 7.7999999999999999e-308 < t

   1. Initial program 33.5%

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

   2. Simplified 33.5%

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

      [Start] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 33.5	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around -inf 1.8%

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

   4. Simplified 1.8%

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

      [Start] 1.8	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \cdot t \]
      mul-1-neg [=>] 1.8	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      distribute-lft-neg-in [=>] 1.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
      distribute-rgt-neg-out [<=] 1.8	\[ \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot \left(-t\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      +-commutative [=>] 1.8	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \cdot t \]
      sub-neg [=>] 1.8	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \cdot t \]
      metadata-eval [=>] 1.8	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \cdot t \]
      +-commutative [=>] 1.8	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \cdot t \]

   5. Applied egg-rr 33.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot t}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot \sqrt{x + -1}\right)} \cdot t \]
      Step-by-step derivation

      [Start] 1.8	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot \left(-t\right)\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}} \cdot t \]
      associate-/r* [=>] 1.8	\[ \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t\right)}}{\sqrt{\frac{x + 1}{-1 + x}}}} \cdot t \]
      sqrt-div [=>] 1.2	\[ \frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t\right)}}{\color{blue}{\frac{\sqrt{x + 1}}{\sqrt{-1 + x}}}} \cdot t \]
      associate-/r/ [=>] 1.3	\[ \color{blue}{\left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \left(-t\right)}}{\sqrt{x + 1}} \cdot \sqrt{-1 + x}\right)} \cdot t \]
      add-sqr-sqrt [=>] 0.0	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \color{blue}{\left(\sqrt{-t} \cdot \sqrt{-t}\right)}}}{\sqrt{x + 1}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      sqrt-unprod [=>] 25.0	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}}}{\sqrt{x + 1}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      sqr-neg [=>] 25.0	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\color{blue}{t \cdot t}}}}{\sqrt{x + 1}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      sqrt-unprod [<=] 33.5	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}}}{\sqrt{x + 1}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      add-sqr-sqrt [<=] 33.5	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot \color{blue}{t}}}{\sqrt{x + 1}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      +-commutative [=>] 33.5	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot t}}{\sqrt{\color{blue}{1 + x}}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      add-sqr-sqrt [=>] 33.5	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot t}}{\sqrt{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      hypot-1-def [=>] 33.5	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot t}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \cdot \sqrt{-1 + x}\right) \cdot t \]
      +-commutative [=>] 33.5	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot t}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot \sqrt{\color{blue}{x + -1}}\right) \cdot t \]

   6. Simplified 39.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{-1 + x}}{t}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \cdot t \]
      Step-by-step derivation

      [Start] 33.5	\[ \left(\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot t}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot \sqrt{x + -1}\right) \cdot t \]
      associate-*l/ [=>] 39.0	\[ \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot t} \cdot \sqrt{x + -1}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}} \cdot t \]
      associate-/r* [=>] 39.1	\[ \frac{\color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{t}} \cdot \sqrt{x + -1}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot t \]
      *-inverses [=>] 39.1	\[ \frac{\frac{\color{blue}{1}}{t} \cdot \sqrt{x + -1}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot t \]
      associate-*l/ [=>] 39.0	\[ \frac{\color{blue}{\frac{1 \cdot \sqrt{x + -1}}{t}}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot t \]
      *-lft-identity [=>] 39.0	\[ \frac{\frac{\color{blue}{\sqrt{x + -1}}}{t}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot t \]
      +-commutative [=>] 39.0	\[ \frac{\frac{\sqrt{\color{blue}{-1 + x}}}{t}}{\mathsf{hypot}\left(1, \sqrt{x}\right)} \cdot t \]

   7. Taylor expanded in t around 0 78.1%

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

   8. Simplified 78.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{x + 1}}} \]
      Step-by-step derivation

      [Start] 78.1	\[ \sqrt{\frac{x - 1}{1 + x}} \]
      +-commutative [=>] 78.1	\[ \sqrt{\frac{x - 1}{\color{blue}{x + 1}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-308}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	65.5%
Cost	27400

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq 2050000000000:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+123}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(t_2 + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, t \cdot \frac{t}{x}\right), t_2\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	65.6%
Cost	14856

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq 63000000000:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+123}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(\ell \cdot \frac{\ell}{x} + \left(2 \cdot \left(\left(1 + \frac{1}{x}\right) \cdot \left(t \cdot t\right)\right) + \frac{\ell}{\frac{x}{\ell}}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	65.5%
Cost	14344

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq 2060000000000:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+123}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left(2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	73.5%
Cost	13768

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq 3.8 \cdot 10^{-166}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	73.4%
Cost	13640

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq 6.5 \cdot 10^{-164}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\ell \cdot \sqrt{\frac{2}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	76.7%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 7
Accuracy	76.3%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 0:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 8
Accuracy	76.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-308}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 9
Accuracy	76.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{x} + \left(-1 - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Accuracy	75.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 0:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11
Accuracy	76.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 0:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Accuracy	75.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq 0:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	39.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023159 
(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)))))