Toniolo and Linder, Equation (7)

Percentage Accurate: 33.8% → 81.6%

Time: 23.8s

Precision: binary64

Cost: 69128

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)\\ t_4 := \frac{t_3}{x}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(t_1 + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{t_3 + t_3}{x \cdot x}\right) + \left(\frac{t_3}{{x}^{3}} + t_4\right)\right)}}\\ \mathbf{elif}\;t \leq -1.42 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_1 + t_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

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 (* 2.0 (+ (* t t) (/ (* t t) x))))
        (t_2 (/ (* l l) x))
        (t_3 (fma (* t 2.0) t (* l l)))
        (t_4 (/ t_3 x)))
   (if (<= t -1.3e+72)
     (+ (/ 1.0 x) -1.0)
     (if (<= t -5.6e-150)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt
          (+
           t_2
           (+
            (fma
             2.0
             (/ (* t t) (pow x 3.0))
             (+ (+ t_1 (/ (* l l) (pow x 3.0))) (/ (+ t_3 t_3) (* x x))))
            (+ (/ t_3 (pow x 3.0)) t_4))))))
       (if (<= t -1.42e-198)
         -1.0
         (if (<= t 9e-17)
           (* t (/ (sqrt 2.0) (sqrt (+ t_2 (+ t_1 t_4)))))
           (+ 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 = 2.0 * ((t * t) + ((t * t) / x));
	double t_2 = (l * l) / x;
	double t_3 = fma((t * 2.0), t, (l * l));
	double t_4 = t_3 / x;
	double tmp;
	if (t <= -1.3e+72) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -5.6e-150) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (fma(2.0, ((t * t) / pow(x, 3.0)), ((t_1 + ((l * l) / pow(x, 3.0))) + ((t_3 + t_3) / (x * x)))) + ((t_3 / pow(x, 3.0)) + t_4)))));
	} else if (t <= -1.42e-198) {
		tmp = -1.0;
	} else if (t <= 9e-17) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (t_1 + t_4))));
	} 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(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = fma(Float64(t * 2.0), t, Float64(l * l))
	t_4 = Float64(t_3 / x)
	tmp = 0.0
	if (t <= -1.3e+72)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= -5.6e-150)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(fma(2.0, Float64(Float64(t * t) / (x ^ 3.0)), Float64(Float64(t_1 + Float64(Float64(l * l) / (x ^ 3.0))) + Float64(Float64(t_3 + t_3) / Float64(x * x)))) + Float64(Float64(t_3 / (x ^ 3.0)) + t_4))))));
	elseif (t <= -1.42e-198)
		tmp = -1.0;
	elseif (t <= 9e-17)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_1 + t_4)))));
	else
		tmp = Float64(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[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * 2.0), $MachinePrecision] * t + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / x), $MachinePrecision]}, If[LessEqual[t, -1.3e+72], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -5.6e-150], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[(N[(l * l), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.42e-198], -1.0, If[LessEqual[t, 9e-17], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$1 + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.29999999999999991e72

   1. Initial program 27.0%

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

   2. Simplified 27.0%

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

      [Start] 27.0%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 27.0%	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 27.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 27.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 27.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 27.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 27.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 27.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around -inf 92.6%

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

   4. Simplified 92.6%

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

      [Start] 92.6%	\[ \sqrt{2} \cdot \frac{t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      associate-*r* [=>] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [=>] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      neg-mul-1 [<=] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      distribute-rgt-neg-in [=>] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      +-commutative [<=] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]
      sub-neg [=>] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 92.6%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   5. Taylor expanded in x around inf 92.7%

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

   if -1.29999999999999991e72 < t < -5.59999999999999993e-150

   1. Initial program 55.1%

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

   2. Simplified 55.3%

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

      [Start] 55.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 55.3%	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around -inf 88.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right)\right) - \left(-1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}} + -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

   4. Simplified 88.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right) - \frac{\left(-\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)\right) - \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x \cdot x}\right) - \left(\frac{-\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x} - \frac{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{{x}^{3}}\right)\right)}}} \cdot t \]
      Step-by-step derivation

      [Start] 88.7%	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right)\right) - \left(-1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}} + -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      associate--l+ [=>] 88.7%	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right) - \left(-1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}} + -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)\right)}}} \cdot t \]
      unpow2 [=>] 88.7%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right) - \left(-1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}} + -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]

   if -5.59999999999999993e-150 < t < -1.42000000000000001e-198

   1. Initial program 30.0%

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

   2. Simplified 30.1%

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

      [Start] 30.0%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 30.1%	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 30.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 30.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 30.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 30.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 30.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 30.1%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Taylor expanded in t around -inf 99.3%

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

   4. Simplified 99.3%

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

      [Start] 99.3%	\[ \sqrt{2} \cdot \frac{t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      associate-*r* [=>] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot t\right)\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [=>] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\left(-1 \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      neg-mul-1 [<=] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\color{blue}{\left(-t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      distribute-rgt-neg-in [=>] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \left(-\sqrt{2}\right)\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      +-commutative [<=] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]
      sub-neg [=>] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 99.3%	\[ \sqrt{2} \cdot \frac{t}{\left(t \cdot \left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   5. Taylor expanded in x around inf 100.0%

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

   if -1.42000000000000001e-198 < t < 8.99999999999999957e-17

   1. Initial program 19.1%

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

   2. Simplified 19.1%

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

      [Start] 19.1%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 19.1%	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 69.1%

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

   4. Simplified 69.1%

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

      [Start] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      associate-*r/ [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]
      mul-1-neg [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]
      +-commutative [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \cdot t \]
      unpow2 [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + {\ell}^{2}\right)}{x}\right)}} \cdot t \]
      associate-*l* [<=] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\left(2 \cdot t\right) \cdot t} + {\ell}^{2}\right)}{x}\right)}} \cdot t \]
      unpow2 [=>] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\left(2 \cdot t\right) \cdot t + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \cdot t \]
      fma-udef [<=] 69.1%	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

   if 8.99999999999999957e-17 < t

   1. Initial program 34.7%

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

   2. Simplified 34.6%

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

      [Start] 34.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 34.6%	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 34.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 34.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 34.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 34.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 34.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 34.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Applied egg-rr 80.6%

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

      [Start] 34.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      associate-*r/ [=>] 34.7%	\[ \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]
      metadata-eval [<=] 34.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{\left(-1\right)}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      sub-neg [<=] 34.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x - 1}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      fma-udef [=>] 34.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, \ell \cdot \left(-\ell\right)\right)}} \]
      +-commutative [<=] 34.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}, \ell \cdot \left(-\ell\right)\right)}} \]
      distribute-rgt-neg-out [=>] 34.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), \color{blue}{-\ell \cdot \ell}\right)}} \]
      fma-neg [<=] 34.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      clear-num [=>] 34.7%	\[ \color{blue}{\frac{1}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2} \cdot t}}} \]

   4. Taylor expanded in l around 0 92.4%

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

   5. Simplified 92.4%

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

      [Start] 92.4%	\[ \frac{1}{\sqrt{\frac{1 + x}{x - 1}}} \]
      +-commutative [=>] 92.4%	\[ \frac{1}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]
      sub-neg [=>] 92.4%	\[ \frac{1}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 92.4%	\[ \frac{1}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

   6. Taylor expanded in x around inf 92.4%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right) + \mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x \cdot x}\right) + \left(\frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{{x}^{3}} + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -1.42 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.6%
Cost	69128

\[\begin{array}{l} t_1 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)\\ t_4 := \frac{t_3}{x}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(t_1 + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{t_3 + t_3}{x \cdot x}\right) + \left(\frac{t_3}{{x}^{3}} + t_4\right)\right)}}\\ \mathbf{elif}\;t \leq -1.42 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_1 + t_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2
Accuracy	81.6%
Cost	35528

\[\begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)\\ t_2 := \frac{t_1}{x}\\ t_3 := \frac{\ell \cdot \ell}{x}\\ t_4 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_3 + \left(\left(t_4 + \frac{t_1 + t_1}{x \cdot x}\right) + t_2\right)}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_3 + \left(t_4 + t_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3
Accuracy	81.6%
Cost	21712

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(t \cdot 2, t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4
Accuracy	76.5%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-198}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell + \ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5
Accuracy	76.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6
Accuracy	75.6%
Cost	452

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

Alternative 7
Accuracy	75.9%
Cost	452

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

Alternative 8
Accuracy	75.2%
Cost	196

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

Alternative 9
Accuracy	38.2%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023165 
(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)))))