Toniolo and Linder, Equation (7)

Percentage Accurate: 20.4% → 81.5%

Time: 32.4s

Precision: binary64

Cost: 87044

• 69.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + 1}{x + -1}\\ t_2 := \sqrt{2} \cdot t\\ t_3 := \frac{t_2}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot t_1 - \ell \cdot \ell}}\\ t_4 := {\ell}^{2} + 2 \cdot {t}^{2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\sqrt{2} \cdot \left(t_4 + t_4\right)}{\sqrt{0.5} \cdot \log \left(e^{x \cdot {t}^{3}}\right)}\right)\\ \mathbf{elif}\;t_3 \leq 1.00000000000001:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{t_1} \cdot \mathsf{hypot}\left(\ell, t_2\right), \ell\right)}{t}}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \end{array} \]

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 (/ (+ x 1.0) (+ x -1.0)))
        (t_2 (* (sqrt 2.0) t))
        (t_3 (/ t_2 (sqrt (- (* (+ (* l l) (* 2.0 (* t t))) t_1) (* l l)))))
        (t_4 (+ (pow l 2.0) (* 2.0 (pow t 2.0)))))
   (if (<= t_3 (- INFINITY))
     (*
      t
      (+
       (/ (* (sqrt 2.0) (sqrt 0.5)) t)
       (*
        -0.125
        (/
         (* (sqrt 2.0) (+ t_4 t_4))
         (* (sqrt 0.5) (log (exp (* x (pow t 3.0)))))))))
     (if (<= t_3 1.00000000000001)
       (/ (sqrt 2.0) (/ (hypot (* (sqrt t_1) (hypot l t_2)) l) t))
       (if (<= t_3 INFINITY)
         t_3
         (+ (+ 1.0 (log1p (expm1 (/ 0.5 (* x x))))) (/ -1.0 x)))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = (x + 1.0) / (x + -1.0);
	double t_2 = sqrt(2.0) * t;
	double t_3 = t_2 / sqrt(((((l * l) + (2.0 * (t * t))) * t_1) - (l * l)));
	double t_4 = pow(l, 2.0) + (2.0 * pow(t, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t * (((sqrt(2.0) * sqrt(0.5)) / t) + (-0.125 * ((sqrt(2.0) * (t_4 + t_4)) / (sqrt(0.5) * log(exp((x * pow(t, 3.0))))))));
	} else if (t_3 <= 1.00000000000001) {
		tmp = sqrt(2.0) / (hypot((sqrt(t_1) * hypot(l, t_2)), l) / t);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (1.0 + log1p(expm1((0.5 / (x * x))))) + (-1.0 / x);
	}
	return tmp;
}

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 = (x + 1.0) / (x + -1.0);
	double t_2 = Math.sqrt(2.0) * t;
	double t_3 = t_2 / Math.sqrt(((((l * l) + (2.0 * (t * t))) * t_1) - (l * l)));
	double t_4 = Math.pow(l, 2.0) + (2.0 * Math.pow(t, 2.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t * (((Math.sqrt(2.0) * Math.sqrt(0.5)) / t) + (-0.125 * ((Math.sqrt(2.0) * (t_4 + t_4)) / (Math.sqrt(0.5) * Math.log(Math.exp((x * Math.pow(t, 3.0))))))));
	} else if (t_3 <= 1.00000000000001) {
		tmp = Math.sqrt(2.0) / (Math.hypot((Math.sqrt(t_1) * Math.hypot(l, t_2)), l) / t);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = (1.0 + Math.log1p(Math.expm1((0.5 / (x * x))))) + (-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 = (x + 1.0) / (x + -1.0)
	t_2 = math.sqrt(2.0) * t
	t_3 = t_2 / math.sqrt(((((l * l) + (2.0 * (t * t))) * t_1) - (l * l)))
	t_4 = math.pow(l, 2.0) + (2.0 * math.pow(t, 2.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t * (((math.sqrt(2.0) * math.sqrt(0.5)) / t) + (-0.125 * ((math.sqrt(2.0) * (t_4 + t_4)) / (math.sqrt(0.5) * math.log(math.exp((x * math.pow(t, 3.0))))))))
	elif t_3 <= 1.00000000000001:
		tmp = math.sqrt(2.0) / (math.hypot((math.sqrt(t_1) * math.hypot(l, t_2)), l) / t)
	elif t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = (1.0 + math.log1p(math.expm1((0.5 / (x * x))))) + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	t_2 = Float64(sqrt(2.0) * t)
	t_3 = Float64(t_2 / sqrt(Float64(Float64(Float64(Float64(l * l) + Float64(2.0 * Float64(t * t))) * t_1) - Float64(l * l))))
	t_4 = Float64((l ^ 2.0) + Float64(2.0 * (t ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(t * Float64(Float64(Float64(sqrt(2.0) * sqrt(0.5)) / t) + Float64(-0.125 * Float64(Float64(sqrt(2.0) * Float64(t_4 + t_4)) / Float64(sqrt(0.5) * log(exp(Float64(x * (t ^ 3.0)))))))));
	elseif (t_3 <= 1.00000000000001)
		tmp = Float64(sqrt(2.0) / Float64(hypot(Float64(sqrt(t_1) * hypot(l, t_2)), l) / t));
	elseif (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(1.0 + log1p(expm1(Float64(0.5 / Float64(x * x))))) + Float64(-1.0 / x));
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[l, 2.0], $MachinePrecision] + N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(t * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(-0.125 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 + t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[Log[N[Exp[N[(x * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.00000000000001], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[l ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$3, N[(N[(1.0 + N[Log[1 + N[(Exp[N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < -inf.0

   1. Initial program 0.7%

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

   2. Simplified 0.7%

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

      [Start] 0.7%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 0.7%	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 22.6%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) \cdot \sqrt{2}}{\sqrt{0.5} \cdot \left({t}^{3} \cdot x\right)}\right)} \cdot t \]

   4. Applied egg-rr 44.8%

      \[\leadsto \left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) \cdot \sqrt{2}}{\sqrt{0.5} \cdot \color{blue}{\log \left(e^{x \cdot {t}^{3}}\right)}}\right) \cdot t \]
      Step-by-step derivation

      [Start] 22.6%	\[ \left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) \cdot \sqrt{2}}{\sqrt{0.5} \cdot \left({t}^{3} \cdot x\right)}\right) \cdot t \]
      add-log-exp [=>] 44.8%	\[ \left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) \cdot \sqrt{2}}{\sqrt{0.5} \cdot \color{blue}{\log \left(e^{{t}^{3} \cdot x}\right)}}\right) \cdot t \]
      *-commutative [=>] 44.8%	\[ \left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) \cdot \sqrt{2}}{\sqrt{0.5} \cdot \log \left(e^{\color{blue}{x \cdot {t}^{3}}}\right)}\right) \cdot t \]

   if -inf.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < 1.00000000000000999

   1. Initial program 54.6%

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

   2. Simplified 54.6%

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

      [Start] 54.6%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 54.5%	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 54.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 54.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 54.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 54.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 54.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 54.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Applied egg-rr 98.4%

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

      [Start] 54.6%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      clear-num [=>] 54.7%	\[ \sqrt{2} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}{t}}} \]
      div-inv [<=] 54.7%	\[ \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}{t}}} \]

   if 1.00000000000000999 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 76.6%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0%	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 0.0%	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      fma-neg [=>] 0.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]
      sub-neg [=>] 0.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{\color{blue}{x + \left(-1\right)}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 0.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + \color{blue}{-1}}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 0.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}, -\ell \cdot \ell\right)}} \]
      fma-def [=>] 0.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}, -\ell \cdot \ell\right)}} \]
      distribute-rgt-neg-in [=>] 0.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \color{blue}{\ell \cdot \left(-\ell\right)}\right)}} \]

   3. Applied egg-rr 3.7%

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

      [Start] 0.0%	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}} \]
      clear-num [=>] 0.0%	\[ \sqrt{2} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}{t}}} \]
      div-inv [<=] 0.0%	\[ \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}{t}}} \]

   4. Taylor expanded in l around 0 3.1%

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

   5. Taylor expanded in x around inf 42.4%

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

   6. Simplified 42.4%

      \[\leadsto \color{blue}{\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x}} \]
      Step-by-step derivation

      [Start] 42.4%	\[ \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x} \]
      sub-neg [=>] 42.4%	\[ \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{1}{x}\right)} \]
      associate-*r/ [=>] 42.4%	\[ \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) + \left(-\frac{1}{x}\right) \]
      metadata-eval [=>] 42.4%	\[ \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) + \left(-\frac{1}{x}\right) \]
      unpow2 [=>] 42.4%	\[ \left(1 + \frac{0.5}{\color{blue}{x \cdot x}}\right) + \left(-\frac{1}{x}\right) \]
      distribute-neg-frac [=>] 42.4%	\[ \left(1 + \frac{0.5}{x \cdot x}\right) + \color{blue}{\frac{-1}{x}} \]
      metadata-eval [=>] 42.4%	\[ \left(1 + \frac{0.5}{x \cdot x}\right) + \frac{\color{blue}{-1}}{x} \]

   7. Applied egg-rr 85.2%

      \[\leadsto \left(1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)}\right) + \frac{-1}{x} \]
      Step-by-step derivation

      [Start] 42.4%	\[ \left(1 + \frac{0.5}{x \cdot x}\right) + \frac{-1}{x} \]
      log1p-expm1-u [=>] 85.2%	\[ \left(1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)}\right) + \frac{-1}{x} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1} - \ell \cdot \ell}} \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\sqrt{2} \cdot \left(\left({\ell}^{2} + 2 \cdot {t}^{2}\right) + \left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right)}{\sqrt{0.5} \cdot \log \left(e^{x \cdot {t}^{3}}\right)}\right)\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1} - \ell \cdot \ell}} \leq 1.00000000000001:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right), \ell\right)}{t}}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1} - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1} - \ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.5%
Cost	87044

\[\begin{array}{l} t_1 := \frac{x + 1}{x + -1}\\ t_2 := \sqrt{2} \cdot t\\ t_3 := \frac{t_2}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot t_1 - \ell \cdot \ell}}\\ t_4 := {\ell}^{2} + 2 \cdot {t}^{2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2} \cdot \sqrt{0.5}}{t} + -0.125 \cdot \frac{\sqrt{2} \cdot \left(t_4 + t_4\right)}{\sqrt{0.5} \cdot \log \left(e^{x \cdot {t}^{3}}\right)}\right)\\ \mathbf{elif}\;t_3 \leq 1.00000000000001:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{t_1} \cdot \mathsf{hypot}\left(\ell, t_2\right), \ell\right)}{t}}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 2
Accuracy	79.7%
Cost	62024

\[\begin{array}{l} t_1 := \frac{x + 1}{x + -1}\\ t_2 := \sqrt{2} \cdot t\\ t_3 := \sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot t_1 - \ell \cdot \ell}\\ t_4 := \frac{t_2}{t_3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(2 \cdot t\right)}}{t_3}\\ \mathbf{elif}\;t_4 \leq 1.00000000000001:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{t_1} \cdot \mathsf{hypot}\left(\ell, t_2\right), \ell\right)}{t}}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 3
Accuracy	74.7%
Cost	49608

\[\begin{array}{l} t_1 := \frac{x + 1}{x + -1}\\ t_2 := \frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot t_1 - \ell \cdot \ell}}\\ \mathbf{if}\;t_2 \leq 0:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 4
Accuracy	74.6%
Cost	43336

\[\begin{array}{l} t_1 := \frac{\sqrt{2} \cdot t}{\sqrt{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{x + 1}{x + -1} - \ell \cdot \ell}}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \end{array} \]

Alternative 5
Accuracy	65.7%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;x \leq -2.12 \cdot 10^{+77}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+210}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6
Accuracy	63.3%
Cost	13900

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+156}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+210}:\\ \;\;\;\;\left(1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{x \cdot x}\right)\right)\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7
Accuracy	61.4%
Cost	13385

\[\begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{-308} \lor \neg \left(t \leq 1.35 \cdot 10^{-90}\right):\\ \;\;\;\;\frac{-1}{x} + \left(1 + \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{0}}{t}}\\ \end{array} \]

Alternative 8
Accuracy	55.7%
Cost	7872

\[\begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \frac{-1}{x} + \left(1 + \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)}\right) \end{array} \]

Alternative 9
Accuracy	52.1%
Cost	2900

\[\begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ t_2 := 1 + t_1\\ t_3 := \frac{t_2 \cdot t_2 + \frac{-1}{x} \cdot \frac{1}{x}}{t_1 + \left(1 - \frac{-1}{x}\right)}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+159}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{+76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;t_1 + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+209}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10
Accuracy	40.1%
Cost	1104

\[\begin{array}{l} t_1 := 1 + \frac{-1}{x}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+158}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+86}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -0.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9:\\ \;\;\;\;\frac{0.5}{x \cdot x} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 11
Accuracy	43.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+116}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \left(1 + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 12
Accuracy	29.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-265}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	29.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-286}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 14
Accuracy	28.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-255}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	14.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023255 
(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)))))