Toniolo and Linder, Equation (7)

Average Accuracy: 31.6% → 87.0%

Time: 29.9s

Precision: binary64

Cost: 28241

Math

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

↓

\[\begin{array}{l} t_1 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -9.6 \cdot 10^{+85}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-306} \lor \neg \left(t \leq 5.4 \cdot 10^{-164}\right) \land t \leq 10^{+91}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{t_1}, \sqrt{\frac{1}{t_1}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \end{array} \]

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 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x)))))
   (if (<= t -9.6e+85)
     (- (sqrt (/ (+ -1.0 x) (+ x 1.0))))
     (if (or (<= t 1.1e-306) (and (not (<= t 5.4e-164)) (<= t 1e+91)))
       (*
        (sqrt 2.0)
        (/ t (sqrt (* 2.0 (+ (* (* t t) (+ 1.0 (/ 2.0 x))) (/ l (/ x l)))))))
       (*
        t
        (/
         (sqrt 2.0)
         (fma t (sqrt t_1) (* (sqrt (/ 1.0 t_1)) (* (/ l x) (/ l t))))))))))

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 + ((2.0 / x) + (2.0 / x));
	double tmp;
	if (t <= -9.6e+85) {
		tmp = -sqrt(((-1.0 + x) / (x + 1.0)));
	} else if ((t <= 1.1e-306) || (!(t <= 5.4e-164) && (t <= 1e+91))) {
		tmp = sqrt(2.0) * (t / sqrt((2.0 * (((t * t) * (1.0 + (2.0 / x))) + (l / (x / l))))));
	} else {
		tmp = t * (sqrt(2.0) / fma(t, sqrt(t_1), (sqrt((1.0 / t_1)) * ((l / x) * (l / t)))));
	}
	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(2.0 / x) + Float64(2.0 / x)))
	tmp = 0.0
	if (t <= -9.6e+85)
		tmp = Float64(-sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	elseif ((t <= 1.1e-306) || (!(t <= 5.4e-164) && (t <= 1e+91)))
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(2.0 * Float64(Float64(Float64(t * t) * Float64(1.0 + Float64(2.0 / x))) + Float64(l / Float64(x / l)))))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / fma(t, sqrt(t_1), Float64(sqrt(Float64(1.0 / t_1)) * Float64(Float64(l / x) * Float64(l / t))))));
	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[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.6e+85], (-N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[Or[LessEqual[t, 1.1e-306], And[N[Not[LessEqual[t, 5.4e-164]], $MachinePrecision], LessEqual[t, 1e+91]]], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 * N[(N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * N[Sqrt[t$95$1], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.59999999999999986e85

   1. Initial program 22.6%

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

   2. Simplified 22.6%

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

      [Start] 22.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 22.6	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 22.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 22.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 22.7%

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

      [Start] 22.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t \]
      clear-num [=>] 22.6	\[ \color{blue}{\frac{1}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \cdot t \]
      associate-*l/ [=>] 22.6	\[ \color{blue}{\frac{1 \cdot t}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]
      *-un-lft-identity [<=] 22.6	\[ \frac{\color{blue}{t}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}} \]
      sqrt-undiv [=>] 22.7	\[ \frac{t}{\color{blue}{\sqrt{\frac{\frac{x + 1}{x - 1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}}} \]
      sub-neg [=>] 22.7	\[ \frac{t}{\sqrt{\frac{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}} \]
      metadata-eval [=>] 22.7	\[ \frac{t}{\sqrt{\frac{\frac{x + 1}{x + \color{blue}{-1}} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}} \]

   4. Taylor expanded in t around -inf 95.6%

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

   5. Simplified 95.6%

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

      [Start] 95.6	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 95.6	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 95.6	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 95.6	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]
      +-commutative [<=] 95.6	\[ -\sqrt{\frac{x + -1}{\color{blue}{x + 1}}} \]
      +-commutative [=>] 95.6	\[ -\sqrt{\frac{\color{blue}{-1 + x}}{x + 1}} \]

   if -9.59999999999999986e85 < t < 1.10000000000000008e-306 or 5.4000000000000003e-164 < t < 1.00000000000000008e91

   1. Initial program 45.2%

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

   2. Simplified 35.6%

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

      [Start] 45.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 45.1	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 39.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 34.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 34.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 34.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 34.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 34.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 34.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 35.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 76.8%

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

   4. Simplified 76.8%

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

      [Start] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   5. Taylor expanded in t around 0 76.8%

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

   6. Simplified 84.0%

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

      [Start] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(1 + 2 \cdot \frac{1}{x}\right)\right)}} \]
      +-commutative [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} \cdot \left(1 + 2 \cdot \frac{1}{x}\right) + \frac{{\ell}^{2}}{x}\right)}}} \]
      unpow2 [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + 2 \cdot \frac{1}{x}\right) + \frac{{\ell}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + \frac{{\ell}^{2}}{x}\right)}} \]
      metadata-eval [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{\color{blue}{2}}{x}\right) + \frac{{\ell}^{2}}{x}\right)}} \]
      unpow2 [=>] 76.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\color{blue}{\ell \cdot \ell}}{x}\right)}} \]
      associate-/l* [=>] 84.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)}} \]

   if 1.10000000000000008e-306 < t < 5.4000000000000003e-164 or 1.00000000000000008e91 < t

   1. Initial program 16.6%

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

   2. Simplified 16.6%

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

      [Start] 16.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 16.6	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 16.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 16.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 30.3%

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

   4. Simplified 30.3%

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

      [Start] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 30.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around inf 72.8%

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

   6. Simplified 85.5%

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

      [Start] 72.8	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      *-commutative [<=] 72.8	\[ \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      fma-def [=>] 72.8	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      distribute-lft-in [=>] 72.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 72.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 72.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 72.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 72.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 72.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 72.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]

3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{+85}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-306} \lor \neg \left(t \leq 5.4 \cdot 10^{-164}\right) \land t \leq 10^{+91}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \left(1 + \frac{2}{x}\right) + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.2%
Cost	14672

\[\begin{array}{l} t_1 := 1 + \frac{2}{x}\\ t_2 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot t_1 + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+87}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-303}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-165}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{t_1}}\\ \end{array} \]

Alternative 2
Accuracy	81.3%
Cost	8656

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+77}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-165}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \end{array} \]

Alternative 3
Accuracy	76.8%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-144}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-303}:\\ \;\;\;\;t \cdot \sqrt{x \cdot \frac{1}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \end{array} \]

Alternative 4
Accuracy	75.5%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \end{array} \]

Alternative 5
Accuracy	76.1%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-160}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \end{array} \]

Alternative 6
Accuracy	75.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	75.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	38.5%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023135 
(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)))))